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Compression that preserves future computation
Executive assessment
The strongest result in this compression investigation is a small, retunable memory of a dynamical objective, not a general-purpose waveform codec. For a declared family of stable scalar filters, a serialized summary can retain loss and terminal-state queries at time constants chosen after the observations have been discarded from working memory. It also supports chronological composition. This is a useful computational capability with a clear information contract; it is not recovery of the original observations.
Two empirical findings distinguish the viable mechanism from an appealing but weaker alternative. An operator-matched waveform codec failed its predeclared joint size-and-computation admission gate against strong classical controls. A separate objective-memory experiment passed: 802-byte Hermite and Chebyshev summaries retained the declared queries across all 36 confirmation records and three merge versions. Chebyshev was much more accurate; the local Hermite evaluator was faster in the tested implementation. Neither result supports a universal spline advantage.
A subsequent mathematical audit explains why repeated merges need not compound interpolation error. Fixed-node evaluation, including first-derivative jets, preserves the pointwise composition algebra. In exact arithmetic, every merge tree therefore produces the interpolant of the exact whole-history statistics. An analytic-function argument also bounds mean-loss interpolation error independently of history length under bounded observations and a fixed stable one-parameter family. Both statements have important numerical and scope limits.
Finally, a new bounded-input streaming test retained the same class of questions through 1,048,576 samples using 438-byte float32 checkpoints and float64 computation. All four declared precision/representation arms passed. These results establish a reproducible mechanism and its limitations, not a practical or state-of-the-art breakthrough. The highest-value next question is whether compact, composable evidence can support reliable re-estimation of a meaningful physical model when raw telemetry is unavailable. Further scalar interpolation sweeps would not answer that question.
1. The compression objective
Compression can reduce at least four different resources: bytes needed to reconstruct observations, state needed to answer a class of questions, parameters needed to execute a learned model, and work needed to perform a computation. These objectives overlap but are not interchangeable. A summary that answers one fitted loss extremely well may be useless for a later change of target. A compact waveform may still require expensive decompression before every physical calculation. An accurate neural weight approximation may not save latency if its layout is poorly matched to the device.
The ambitious target here is a computational representation of physical evidence: small machines retain or exchange enough information to perform valuable later operations, including some model revisions. The operations, accuracy requirement and admissible revisions must be explicit. A meaningful demonstration would preserve the quality of a scientific or engineering conclusion while making it feasible under a previously binding memory, communication or repeated-computation budget.
This framing does not require compressed sensing, kernels, oblique projection or splines to be the winning method. Those ideas offer distinct mechanisms. Compressed sensing exploits structured signal classes and measurement conditions. Kernel methods often avoid explicitly materializing a larger feature space; they do not generally recover arbitrary information from fewer coordinates. An oblique projection can be compatible with an acquisition or evaluation operator, but its discarded nullspace is still discarded. Hilbert- space structure alone supplies neither a compression ratio nor an application.
The appropriate comparison is therefore an information contract. Which inputs are available to the encoder? Which questions can be asked afterward? Which metadata, learned parameters, numerical arrays and residuals must be retained? How much initial work is required, and when is it amortized? Without these answers, a large raw-byte-to-summary ratio can describe an ordinary statistic rather than a scientific compression advance.
2. Literature and competitive boundaries
Operator matching and sparse innovations
Operator-based signal models connect differential equations, sparse driving events and localized representations. Pad and Unser analyze operator-like wavelets for sparse AR(1) processes using independence/information and estimation criteria. This is strong motivation for a matched transform, but not a theorem that a particular finite-bit AR(2) codec beats a predictive compressor on noisy measurements.[^1] The distinction matters: matching a generative operator can concentrate coefficients, while residual noise, event locations and bitstream overheads still determine an actual rate-distortion result.
Finite-rate-of-innovation sampling supplies a different route. Polynomial or exponential reproduction can make suitable samples encode a finite set of moments, from which a structured innovation stream can be recovered under the model's assumptions. The acquisition kernel, innovation density, boundary conditions and noise sensitivity are essential parts of the method.[^2] Sparse innovations do not mean that arbitrary dense noise has become exactly recoverable from the same small number of measurements.
Exponential-spline multiresolution constructions and Hermite multiwavelets are also established. They make reproduction and cancellation properties useful for algorithms; they should be treated as a foundation rather than claimed as new because they appear inside a learning system.[^3] The present waveform implementation uses a discrete orthogonal exponential-polynomial multiwavelet, not a newly derived cardinal exponential B-spline generator.
Generalized sampling, sketches and inference
Generalized sampling and quasi-projection separate analysis functions from the reconstruction space. They allow approximation and computational properties to be designed jointly, while retaining stability and consistency conditions. Short quasi-dual kernels are one established way to approximate oblique projection efficiently.[^4] These constructions help identify compatible interfaces; they do not eliminate information loss outside a declared space.
Compressed sensing with coherent dictionaries already distinguishes analysis sparsity from orthogonal-basis sparsity and gives recovery results under appropriate restricted-isometry assumptions.[^5] Compressive statistical learning instead retains sketches for a prescribed statistical model or risk. Its guarantees are tied to that model class, not arbitrary future inference.[^6] These are useful precedents for measuring compression by what can still be inferred rather than by exact sample reconstruction.
Approximate sufficient statistics are particularly close prior art. PASS-GLM uses polynomial approximation to enable streaming and distributed inference, with no extra aggregation approximation and explicit statistical guarantees.[^7] Temporal-model parameter inference also has polynomial approximate-statistic constructions, including Erol's extended parameter filter.[^8] The current objective-memory result cannot be presented as the invention of approximate sufficiency, streaming learning from polynomial summaries, or non-compounding aggregation in general.
Scientific codecs and parameterized computation
Modern scientific compression already addresses more than storage. ZFP supports compressed numerical arrays and access patterns useful for computation.[^9] MGARD provides error-controlled multilevel representations; progressive retrieval methods also control derived quantities rather than only primary sample errors.[^10] An operator-aware archive must therefore be compared with these capabilities, not only with uncompressed arrays or a weak uniform quantizer.
Parameter collocation is a classical way to replace repeated parameterized solves with interpolation of deterministic solutions. Sparse-grid work also makes the curse of parameter dimension explicit.[^11] Associative temporal methods compose dynamical maps and quadratic costs, while state-space/attention duality exploits structured linear algebra for efficient sequence computation.[^12] The new implementation uses related algebra, but neither parallel scans nor small matrix recurrences are themselves a novelty claim.
Kernels and neural-memory compression
Exponential and oscillatory kernel structure already enables fast Gaussian- process computations. Celerite is a particularly relevant strong comparator for physical time series, rather than dense cubic-time GP regression.[^13] This suggests a possible bridge to compact inference memories, but the present least-squares experiment is not a GP likelihood or posterior experiment.
Closer precedents further narrow that bridge. Greengard constructs Fourier representations valid across hyperparameter ranges, allowing inference independent of observation count after preprocessing on a specified domain.[^14] Dong et al. already use numerical surrogates for GP kernel-learning quantities, and associative Bayesian filtering has established Gaussian specializations.[^15] These are mandatory comparisons, not evidence that a generic compressed-GP experiment would be a new capability.
The broader neural-inference opportunity is large but less directly supported by these results. Performer uses structured random features to approximate attention kernels.[^16] Query-agnostic cache compression also has dedicated methods such as KVzip.[^17] Lower-bound research shows why unrestricted future attention queries cannot simply be assumed to admit tiny universal summaries; the precise dimensional and structural assumptions matter.[^18] Smooth one-dimensional filter-parameter interpolation does not resolve that problem.
3. The role of the spline toolbox
The toolbox contributes most when it makes an operation explicit. Reproduction properties identify signal components that a basis can represent without approximation in an ideal model. Annihilating operators can turn smooth structured signals into sparse innovations. Local support and cardinal structure allow predictable evaluation and small fixed transforms. Inner- product calculus permits exact or well-controlled quadratic calculations without dense resampling. State realizations can execute long homogeneous segments with compact transition operators.
Each advantage has a corresponding boundary. Reproduction is not finite-bit exactness after quantization. Local support is not an unconditional guarantee against forgetting when shared parameters, coordinates or model structure change. A matrix-vector formulation is not automatically a GPU speedup after indexing, memory traffic and setup are charged. Exact Gram statistics preserve a declared feature space; they do not create missing cross-statistics when a genuinely new feature is introduced.
The compression investigation makes a further distinction between operations that are algebraically compatible with a representation and operations that merely admit a small approximation error once. This is why the merge question is more consequential than another isolated evaluator timing. A representation that preserves the defining data of a composition can avoid a new interpolation error at every aggregation step, even though it remains approximate between its defining nodes.
4. Waveform-to-program experiment
The first experiment asked whether a matched representation could make a physical waveform both smaller and cheaper to reuse computationally. Signals were encoded using an operator-matched exponential-polynomial multiwavelet, its polynomial counterpart, Haar, DCT, closed-loop double-pole prediction and direct quantization. Query workloads evaluated low-pass responses and energies. Applicable controls received the same compiled event/state execution route; all controls also had an optimized dense reference route.
The confirmation panel contained 48 records in six independent seed groups: two lengths, four signal families, three distortion ceilings and six methods, giving 864 arm checks. The families separated matched sparse events, pole mismatch, dense stochastic forcing and measurement noise. Actual serialized bytes included method metadata and coefficient/index coding. Encoding, decoding, compilation and repeated-query costs were recorded. All numerical checks passed.
At N=65,536 and a one-percent RMS-relative error ceiling on matched records, the following medians illustrate the tradeoff. Achieved distortions differ; this is a common-ceiling comparison, not an interpolated equal-distortion rate-distortion curve.
| Method | Serialized bytes | Encode-to-272-query cost |
|---|
| Matched operator | 358 | 5.81 ms |
| Polynomial multiwavelet | 548 | 8.26 ms |
| Haar | 1,029.5 | 18.88 ms |
| DCT | 634 | 18.11 ms |
| Closed-loop prediction | 428 | 24.29 ms |
| Direct quantization | 1,989 | 47.08 ms |
The favorable matched example did not satisfy the predeclared joint gate: at least five of six seeds had to achieve both a byte ratio at most 0.8 and a total-cost ratio at most 0.5 against the best competing route. Zero of six seeds met both conditions. Median paired ratios were 0.812 for bytes and 0.694 for total cost. Dense forcing favored DCT on size, and measurement noise largely removed the matched representation's storage advantage.
The conclusion is a closed negative application-admission result, with useful engineering retained. It is not evidence that exponential splines are generally ineffective, nor is it permission to tune the same panel until its gate passes. The experiment also did not benchmark the full set of modern scientific codecs. Its reproducible record is FINDINGS_01.md and SUMMARY_01.json.
5. Retunable objective memory
The second experiment changed the information contract, not the parameters of the failed waveform test. For a sampled stable filter, zn=qzn−1+(1−q)xn with q=exp(−Δt/τ), it retained three functions of log(τ): the zero-initial- state terminal response B, an initial-state/residual cross-term H/N, and the zero-state mean squared loss R/N. Deterministic quantities A=qN and P=∑i=1Nq2i depend only on the model and sample count.
For incoming state s, terminal state is As+B and mean loss is (P/N)s2+2(H/N)s+R/N. Thus an unspecified incoming state is handled by exact affine/quadratic algebra, while an unspecified time constant is handled by interpolation over the declared interval [0.002,0.2] seconds. The encoder knows the interval and model family; it does not know the eventual random queries. It has already observed the target values used in the loss.
Four representations shared two equal scalar budgets: ordinary linear tables, natural cubic tables, cardinal cubic Hermite values plus analytic derivatives, and Chebyshev-Lobatto interpolation. Hermite used half as many nodes so that derivatives were not free extra stored data. The 36-record confirmation panel used six seeds, two lengths and three target regimes, including model misspecification. Every record was encoded directly and as 32 blocks merged sequentially and in a balanced tree.
| Representation | Bytes | Passing versions / 108 | Worst mean-loss error |
|---|
| Linear, 32 scalars/function | 802 | 0 | 3.64e-4 |
| Linear, 64 | 1,570 | 0 | 8.80e-5 |
| Natural cubic, 32 | 802 | 45 | 9.70e-5 |
| Natural cubic, 64 | 1,570 | 108 | 2.26e-5 |
| Cardinal Hermite, 32 | 802 | 108 | 7.88e-6 |
| Cardinal Hermite, 64 | 1,570 | 108 | 4.17e-7 |
| Chebyshev, 32 | 802 | 108 | 4.60e-11 |
| Chebyshev, 64 | 1,570 | 108 | 2.27e-14 |
Passing also required terminal error and optimizer regret at most 1e-4, nonnegative quadratic form within tolerance, successful optimization and a serialized object no larger than 2,048 bytes. Mean-loss error alone therefore does not explain every failure in the table. Both Hermite budgets, both Chebyshev budgets and the larger natural cubic table passed every capability check and the computational gate. No materially indefinite quadratic was observed at the tested queries.
For 259 warm queries at N=65,536, median paired speedups over the compiled raw-data reference were approximately 634x for Hermite-32 and 218x for Chebyshev-32. Whole-record encoding amortized after median 0.075 and 0.126 such query batches respectively. These are implementation-specific repeated- query savings, not speedups over the best possible competing summary, and not evidence that preprocessing is free for a single question. All block encoding and merge costs remain in the complete records.
The comparison is scientifically useful precisely because it does not force a spline win. Hermite offers fast local evaluation and a smaller-budget advantage over an ordinary natural cubic table. Chebyshev exploits the analytic response functions much more accurately. Future implementations may change their timing order. A useful computational representation should select the appropriate approximation family instead of treating basis identity as its success criterion.
6. Composition and the history-length bound
At fixed nodes, evaluation preserves sums and pointwise products. Reconstructing an interpolant and evaluating it again at those nodes returns the same values. The block-merge formulas depend only on these values and deterministic metadata. Consequently, interpolating after each merge gives the same nodal statistics as computing the exact merged functions first and then interpolating once. Uniqueness of the interpolant makes the final functions agree in real arithmetic. Chronological order remains essential; only parenthesization is interchangeable.
Hermite data obeys the analogous first-jet algebra: the derivative of a product uses the product rule. This provides the same composition property when value and derivative nodes remain fixed. It is not a property of arbitrary oblique or orthogonal projection. For example, L2 projection onto span{1,u} on [-1,1] gives P(u2)=1/3, so P(P(u2)P(u))=u/3, whereas P(u3)=3u/5. That counterexample disproves the required commutation identity for a generic projection.
For the stable single-pole family, analytic continuation supplies a useful uniform bound. With bounded inputs and targets, a suitable complex ellipse in log(tau) retains a uniformly stable filter. Its response, normalized cross- term and normalized loss are bounded independently of the number of samples. The classical Chebyshev analytic-function theorem then gives geometric decay of interpolation error with degree.[^19] The full derivation, including a fixed ellipse parameter and amplitude factors, is in THEORY_03.md.
The exposed-record audit found node-value and Hermite-derivative discrepancies below 2.60e-14. All observed Chebyshev errors lay below the derived real- arithmetic bounds. This is a post-exposure explanation, not another independent confirmation panel. Floating-point node evaluations, transforms and merges are not formally certified. The mean-loss bound does not give an N-independent absolute total-loss or posterior bound. A fixed amplitude envelope and fixed one-dimensional query family are essential assumptions.
7. Long-stream finite checkpoints
A separate numerical scope test encoded bounded observations in chunks of 1,024 samples, restoring the current summary from a quantized checkpoint before each subsequent merge. It examined both interpolation families with float64 and float32 checkpoint payloads, while keeping computation in float64. Twelve fresh records used six independent seeds, constant or switching target dynamics, and prefixes from 1,024 to 1,048,576 samples. Queries were selected only after all checkpoints were saved.
| Checkpoint payload | Bytes | Worst terminal error | Worst mean-loss error |
|---|
| Chebyshev, float64 | 822 | 3.69e-10 | 9.37e-12 |
| Chebyshev, float32 | 438 | 2.03e-8 | 1.30e-8 |
| Hermite, float64 | 822 | 1.95e-5 | 4.42e-6 |
| Hermite, float32 | 438 | 1.96e-5 | 4.43e-6 |
All four arms passed all 48 prefix/record checks. The changed header includes known amplitude envelopes and a larger length field, so the byte counts are not a revision of the preceding experiment's format. Independent SciPy statistics checks agreed with the compiled reference to within 1.45e-15. The largest stream represents about 17.5 minutes at the stated sampling rate, not an indefinite-duration test.
This closes a specific implementation gap: input working chunks and retained summaries need not grow with the tested history. The validation phase still regenerated raw observations to check the claims, and the full process used up to 180.6 MiB. The first encoding phase's process high-water mark was 106.4 MiB, dominated by the research environment rather than the serialized object. Neither number is an embedded-device power or isolated allocator measurement.
Preserving an old average objective under target drift also does not establish a good adaptation policy. A learner may need recency weighting, change-point reasoning or a revised loss, none of which can be recovered from missing statistics by naming the object a memory. The result concerns faithful queries inside the fixed contract, not unrestricted continual learning.
8. High-impact application assessment
The closest supported vision is revisable scientific telemetry: an instrument can transmit a small evidence object rather than a long recording, and a recipient can later reconsider a meaningful physical model. The value would come from preserving scientific conclusions under a communication or memory limit, not from displaying a large reduction ratio for a single statistic. Noise-aware inference and uncertainty are important because many consequential decisions depend on distinguishing competing dynamics rather than merely minimizing a scalar loss accurately.
| Vision | Decisive capability | Mandatory comparison |
|---|
| Revisable telemetry | Later physical re-estimation with reliable uncertainty | State-space/semiseparable inference, fixed parameter banks and compressed raw evidence |
| Portable machine experience | Useful learning after episodes are exchanged | Fixed-feature Gram memory, recursive identification and local replay |
| Computational simulation archive | Later operators and derived quantities remain accurate | MGARD/ZFP, predictive codecs and reduced models |
| Compressed neural inference | Future sequence/model decisions survive smaller state | Attention/cache methods, quantization and distillation |
These are alternative destinations, not interchangeable names for the current scalar mechanism. The failed waveform experiment does not admit a simulation- archive claim. The passed objective experiment does not admit a neural-cache claim. A small state on a laptop does not establish edge-device energy savings. The role of a vision is to select consequential tests, not to convert every local improvement into evidence for the entire vision.
A tempting next step would add many first-order filters after compression and claim architectural growth. That is insufficient on its own. If the new filter trajectories are interpolated from K retained node trajectories, they remain in the same K-dimensional span. Their quadratic losses can be represented by an ordinary Gram matrix and cross-vector. A free linear readout of the fixed bank contains every such interpolated mixture and is the mandatory control. Changing a pole's name or the mixture width does not create new evidence.
The more demanding question is whether a compact family of composable physical inference objects retains enough flexibility for an application that existing summaries cannot serve as well. Linear-Gaussian block factors are one plausible bridge: fixed-parameter elimination is structured, but parameterized likelihoods introduce normalization, conditioning and uncertainty requirements absent from the current least-squares test. This is a prospective hypothesis, not an implemented result or an established novelty claim.
In particular, observation-count independence on a fixed spatial domain must not be confused with independence of a stream's increasing duration. A future application must separate those axes. The current scalar mean-loss result does not prove a duration-independent posterior representation. The additional competitive assessment in NEXT_COMPARATOR_AUDIT.md therefore does not admit a generic Gaussian-inference panel without a sharper capability requirement.
9. Next experiment and stopping criteria
The next admitted application should involve noisy physical time series, meaningful competing parameter explanations and a hard retention or transmission budget. Before numerical comparison, it needs a precise target inference, timestamp and missingness semantics, parameter/noise domain and required accuracy of the conclusion. Posterior or likelihood-ratio tasks must control the relevant total error; a small per-sample discrepancy can become important after multiplication by a long history.
The reference should exploit the same physical structure. For exponential kernel families, this means state-space or semiseparable computation rather than an unnecessarily dense solver. A plain parameter bank with the same interpolation and byte accounting is also required. If it reproduces the proposal, novelty must lie in a demonstrably different capability, guarantee or resource tradeoff, not terminology. Ingestion, encoding, query costs and any recoverable residual evidence all belong in the comparison.
A convincing outcome would allow the recipient to make several later-selected scientific inferences at near-reference quality while using materially less transmission, retention and repeated work than those controls. Episodes or instruments would remain grouped in evaluation, with independent held-out cases and all declared query families reported. A failure of uncertainty or coverage would remain a failure even if a headline average error were small.
The present scalar screens are closed. No additional degree sweep, easier target, optimizer search or dataset subset is justified by them. The evidence supports a concrete direction and a reusable implementation, but the remaining gap is application-level utility and novelty. A high-impact compression result would emerge from crossing that gap, not from making an already accurate one-parameter surrogate marginally smaller.
10. Evidence package and scope
The research package contains separate protocols, complete numerical records, actual serialized objects, source hashes, analysis scripts and tests. Study 01 has 864 checked codec outputs; study 02 has 10,080 checked block/final memory objects; the algebra audit is explicitly post-exposure; study 04 has 192 checked long-stream checkpoint objects. Repeated queries, merge orders and related target regimes are not treated as independent experimental subjects.
The numerical work uses one process/thread and the existing environment. No reserved trading data, production systems or neural training checkpoints are involved. The tested compression mechanisms are synthetic and deliberately bounded. Results and failures are preserved without reopening earlier panels. Reproduction instructions and artifact accounting are provided in README.md; reading depth and prior-art limits are recorded separately in READING_LEDGER.md.
Sources
[^1]: Pedram Pad and Michael Unser. Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes. IEEE Transactions on Signal Processing, 2015. DOI 10.1109/TSP.2015.2447494. [^2]: Pier Luigi Dragotti, Martin Vetterli and Thierry Blu. Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang-Fix. IEEE Transactions on Signal Processing, 2007. [^3]: Ildar Khalidov and Michael Unser. From Differential Equations to the Construction of New Wavelet-Like Bases. Preprint associated with the 2006 publication. Mariantonia Cotronei and Nada Sissouno. A note on Hermite multiwavelets with polynomial and exponential vanishing moments, 2017. [^4]: Michael Unser. Quasi-Orthogonality and Quasi-Projections. Applied and Computational Harmonic Analysis 3, 201-214, 1996. Author abstract on quasi-projection and short quasi-duals. [^5]: Emmanuel Candes, Yonina Eldar, Deanna Needell and Paige Randall. Compressed sensing with coherent and redundant dictionaries. 2010 preprint. [^6]: Remi Gribonval, Gilles Blanchard, Nicolas Keriven and Yann Traonmilin. Compressive Statistical Learning with Random Feature Moments. 2017 preprint and subsequent revisions. [^7]: Jonathan H. Huggins, Ryan P. Adams and Tamara Broderick. PASS-GLM: polynomial approximate sufficient statistics for scalable Bayesian GLM inference. NeurIPS, 2017. [^8]: Yusuf Bugra Erol. Joint State and Parameter Estimation in Temporal Models. UC Berkeley technical report, 2018; chapter 3, particularly sections 3.3-3.5. [^9]: Lawrence Livermore National Laboratory. ZFP project. Compressed floating-point arrays and access/computation documentation; accessed September 2026. [^10]: MGARD authors. MGARD: A multigrid framework for high-performance, error-controlled data compression and refactoring, 2024. Xuan Wu et al. Error-controlled Progressive Retrieval of Scientific Data under Derivable Quantities of Interest, SC 2024. [^11]: Fabio Nobile, Raul Tempone and Clayton G. Webster. A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. Author preprint dated June 22, 2006; related journal publication 2008. [^12]: Temporal Parallelisation of the HJB Equation and Continuous-Time Linear Quadratic Control, 2022 preprint. Tri Dao and Albert Gu. Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality, 2024. [^13]: Daniel Foreman-Mackey et al. Fast and scalable Gaussian process modeling with applications to astronomical time series. 2017. Celerite kernel documentation supplies the exponential/oscillatory kernel interface. [^14]: Philip Greengard. Efficient Fourier representations of families of Gaussian processes. 2021 preprint, inspected revision dated May 31, 2024; sections 2-3 and Algorithm 1. [^15]: Kun Dong et al. Scalable Log Determinants for Gaussian Process Kernel Learning. NeurIPS, 2017; sections 3.2 and 3.5. Simo Sarkka and Angel F. Garcia-Fernandez. Temporal Parallelization of Bayesian Smoothers. 2019 preprint. [^16]: Krzysztof Choromanski et al. Rethinking Attention with Performers. 2020 preprint; ICLR 2021. [^17]: KVzip: Query-Agnostic KV Cache Compression with Context Reconstruction. 2025 preprint. Screened as competitive context, not benchmarked here. [^18]: Themistoklis Haris and Krzysztof Onak. Compression Barriers for Autoregressive Transformers. 2025 preprint. Its stated lower bounds depend on the attention/query and dimension assumptions. [^19]: Lloyd N. Trefethen. Approximation Theory and Approximation Practice, chapter 8 source. Theorem 8.2, equation (8.3), Chebyshev interpolation of analytic functions.
View raw MD source
# Compression that preserves future computation
## Executive assessment
The strongest result in this compression investigation is a small, retunable
memory of a dynamical objective, not a general-purpose waveform codec. For a
declared family of stable scalar filters, a serialized summary can retain loss
and terminal-state queries at time constants chosen after the observations
have been discarded from working memory. It also supports chronological
composition. This is a useful computational capability with a clear information
contract; it is not recovery of the original observations.
Two empirical findings distinguish the viable mechanism from an appealing but
weaker alternative. An operator-matched waveform codec failed its predeclared
joint size-and-computation admission gate against strong classical controls.
A separate objective-memory experiment passed: 802-byte Hermite and Chebyshev
summaries retained the declared queries across all 36 confirmation records and
three merge versions. Chebyshev was much more accurate; the local Hermite
evaluator was faster in the tested implementation. Neither result supports a
universal spline advantage.
A subsequent mathematical audit explains why repeated merges need not compound
interpolation error. Fixed-node evaluation, including first-derivative jets,
preserves the pointwise composition algebra. In exact arithmetic, every merge
tree therefore produces the interpolant of the exact whole-history statistics.
An analytic-function argument also bounds mean-loss interpolation error
independently of history length under bounded observations and a fixed stable
one-parameter family. Both statements have important numerical and scope limits.
Finally, a new bounded-input streaming test retained the same class of questions
through 1,048,576 samples using 438-byte float32 checkpoints and float64
computation. All four declared precision/representation arms passed. These
results establish a reproducible mechanism and its limitations, not a practical
or state-of-the-art breakthrough. The highest-value next question is whether
compact, composable evidence can support reliable re-estimation of a meaningful
physical model when raw telemetry is unavailable. Further scalar interpolation
sweeps would not answer that question.
## 1. The compression objective
Compression can reduce at least four different resources: bytes needed to
reconstruct observations, state needed to answer a class of questions, parameters
needed to execute a learned model, and work needed to perform a computation.
These objectives overlap but are not interchangeable. A summary that answers
one fitted loss extremely well may be useless for a later change of target.
A compact waveform may still require expensive decompression before every
physical calculation. An accurate neural weight approximation may not save
latency if its layout is poorly matched to the device.
The ambitious target here is a computational representation of physical
evidence: small machines retain or exchange enough information to perform
valuable later operations, including some model revisions. The operations,
accuracy requirement and admissible revisions must be explicit. A meaningful
demonstration would preserve the quality of a scientific or engineering
conclusion while making it feasible under a previously binding memory,
communication or repeated-computation budget.
This framing does not require compressed sensing, kernels, oblique projection
or splines to be the winning method. Those ideas offer distinct mechanisms.
Compressed sensing exploits structured signal classes and measurement
conditions. Kernel methods often avoid explicitly materializing a larger
feature space; they do not generally recover arbitrary information from fewer
coordinates. An oblique projection can be compatible with an acquisition or
evaluation operator, but its discarded nullspace is still discarded. Hilbert-
space structure alone supplies neither a compression ratio nor an application.
The appropriate comparison is therefore an information contract. Which inputs
are available to the encoder? Which questions can be asked afterward? Which
metadata, learned parameters, numerical arrays and residuals must be retained?
How much initial work is required, and when is it amortized? Without these
answers, a large raw-byte-to-summary ratio can describe an ordinary statistic
rather than a scientific compression advance.
## 2. Literature and competitive boundaries
### Operator matching and sparse innovations
Operator-based signal models connect differential equations, sparse driving
events and localized representations. Pad and Unser analyze operator-like
wavelets for sparse AR(1) processes using independence/information and estimation
criteria. This is strong motivation for a matched transform, but not a theorem
that a particular finite-bit AR(2) codec beats a predictive compressor on noisy
measurements.[^1] The distinction matters: matching a generative operator can
concentrate coefficients, while residual noise, event locations and bitstream
overheads still determine an actual rate-distortion result.
Finite-rate-of-innovation sampling supplies a different route. Polynomial or
exponential reproduction can make suitable samples encode a finite set of
moments, from which a structured innovation stream can be recovered under the
model's assumptions. The acquisition kernel, innovation density, boundary
conditions and noise sensitivity are essential parts of the method.[^2]
Sparse innovations do not mean that arbitrary dense noise has become exactly
recoverable from the same small number of measurements.
Exponential-spline multiresolution constructions and Hermite multiwavelets are
also established. They make reproduction and cancellation properties useful
for algorithms; they should be treated as a foundation rather than claimed as
new because they appear inside a learning system.[^3] The present waveform
implementation uses a discrete orthogonal exponential-polynomial multiwavelet,
not a newly derived cardinal exponential B-spline generator.
### Generalized sampling, sketches and inference
Generalized sampling and quasi-projection separate analysis functions from the
reconstruction space. They allow approximation and computational properties
to be designed jointly, while retaining stability and consistency conditions.
Short quasi-dual kernels are one established way to approximate oblique
projection efficiently.[^4] These constructions help identify compatible
interfaces; they do not eliminate information loss outside a declared space.
Compressed sensing with coherent dictionaries already distinguishes analysis
sparsity from orthogonal-basis sparsity and gives recovery results under
appropriate restricted-isometry assumptions.[^5] Compressive statistical
learning instead retains sketches for a prescribed statistical model or risk.
Its guarantees are tied to that model class, not arbitrary future inference.[^6]
These are useful precedents for measuring compression by what can still be
inferred rather than by exact sample reconstruction.
Approximate sufficient statistics are particularly close prior art. PASS-GLM
uses polynomial approximation to enable streaming and distributed inference,
with no extra aggregation approximation and explicit statistical guarantees.[^7]
Temporal-model parameter inference also has polynomial approximate-statistic
constructions, including Erol's extended parameter filter.[^8] The current
objective-memory result cannot be presented as the invention of approximate
sufficiency, streaming learning from polynomial summaries, or non-compounding
aggregation in general.
### Scientific codecs and parameterized computation
Modern scientific compression already addresses more than storage. ZFP supports
compressed numerical arrays and access patterns useful for computation.[^9]
MGARD provides error-controlled multilevel representations; progressive
retrieval methods also control derived quantities rather than only primary
sample errors.[^10] An operator-aware archive must therefore be compared with
these capabilities, not only with uncompressed arrays or a weak uniform
quantizer.
Parameter collocation is a classical way to replace repeated parameterized
solves with interpolation of deterministic solutions. Sparse-grid work also
makes the curse of parameter dimension explicit.[^11] Associative temporal
methods compose dynamical maps and quadratic costs, while state-space/attention
duality exploits structured linear algebra for efficient sequence computation.[^12]
The new implementation uses related algebra, but neither parallel scans nor
small matrix recurrences are themselves a novelty claim.
### Kernels and neural-memory compression
Exponential and oscillatory kernel structure already enables fast Gaussian-
process computations. Celerite is a particularly relevant strong comparator
for physical time series, rather than dense cubic-time GP regression.[^13]
This suggests a possible bridge to compact inference memories, but the present
least-squares experiment is not a GP likelihood or posterior experiment.
Closer precedents further narrow that bridge. Greengard constructs Fourier
representations valid across hyperparameter ranges, allowing inference
independent of observation count after preprocessing on a specified domain.[^14]
Dong et al. already use numerical surrogates for GP kernel-learning quantities,
and associative Bayesian filtering has established Gaussian specializations.[^15]
These are mandatory comparisons, not evidence that a generic compressed-GP
experiment would be a new capability.
The broader neural-inference opportunity is large but less directly supported
by these results. Performer uses structured random features to approximate
attention kernels.[^16] Query-agnostic cache compression also has dedicated
methods such as KVzip.[^17] Lower-bound research shows why unrestricted future
attention queries cannot simply be assumed to admit tiny universal summaries;
the precise dimensional and structural assumptions matter.[^18] Smooth
one-dimensional filter-parameter interpolation does not resolve that problem.
## 3. The role of the spline toolbox
The toolbox contributes most when it makes an operation explicit. Reproduction
properties identify signal components that a basis can represent without
approximation in an ideal model. Annihilating operators can turn smooth
structured signals into sparse innovations. Local support and cardinal
structure allow predictable evaluation and small fixed transforms. Inner-
product calculus permits exact or well-controlled quadratic calculations
without dense resampling. State realizations can execute long homogeneous
segments with compact transition operators.
Each advantage has a corresponding boundary. Reproduction is not finite-bit
exactness after quantization. Local support is not an unconditional guarantee
against forgetting when shared parameters, coordinates or model structure
change. A matrix-vector formulation is not automatically a GPU speedup after
indexing, memory traffic and setup are charged. Exact Gram statistics preserve
a declared feature space; they do not create missing cross-statistics when a
genuinely new feature is introduced.
The compression investigation makes a further distinction between operations
that are algebraically compatible with a representation and operations that
merely admit a small approximation error once. This is why the merge question
is more consequential than another isolated evaluator timing. A representation
that preserves the defining data of a composition can avoid a new interpolation
error at every aggregation step, even though it remains approximate between
its defining nodes.
## 4. Waveform-to-program experiment
The first experiment asked whether a matched representation could make a
physical waveform both smaller and cheaper to reuse computationally. Signals
were encoded using an operator-matched exponential-polynomial multiwavelet,
its polynomial counterpart, Haar, DCT, closed-loop double-pole prediction and
direct quantization. Query workloads evaluated low-pass responses and energies.
Applicable controls received the same compiled event/state execution route;
all controls also had an optimized dense reference route.
The confirmation panel contained 48 records in six independent seed groups:
two lengths, four signal families, three distortion ceilings and six methods,
giving 864 arm checks. The families separated matched sparse events, pole
mismatch, dense stochastic forcing and measurement noise. Actual serialized
bytes included method metadata and coefficient/index coding. Encoding, decoding,
compilation and repeated-query costs were recorded. All numerical checks passed.
At N=65,536 and a one-percent RMS-relative error ceiling on matched records,
the following medians illustrate the tradeoff. Achieved distortions differ;
this is a common-ceiling comparison, not an interpolated equal-distortion
rate-distortion curve.
| Method | Serialized bytes | Encode-to-272-query cost |
|---|---:|---:|
| Matched operator | 358 | 5.81 ms |
| Polynomial multiwavelet | 548 | 8.26 ms |
| Haar | 1,029.5 | 18.88 ms |
| DCT | 634 | 18.11 ms |
| Closed-loop prediction | 428 | 24.29 ms |
| Direct quantization | 1,989 | 47.08 ms |
The favorable matched example did not satisfy the predeclared joint gate:
at least five of six seeds had to achieve both a byte ratio at most 0.8 and a
total-cost ratio at most 0.5 against the best competing route. Zero of six
seeds met both conditions. Median paired ratios were 0.812 for bytes and 0.694
for total cost. Dense forcing favored DCT on size, and measurement noise
largely removed the matched representation's storage advantage.
The conclusion is a closed negative application-admission result, with useful
engineering retained. It is not evidence that exponential splines are generally
ineffective, nor is it permission to tune the same panel until its gate passes.
The experiment also did not benchmark the full set of modern scientific codecs.
Its reproducible record is `FINDINGS_01.md` and `SUMMARY_01.json`.
## 5. Retunable objective memory
The second experiment changed the information contract, not the parameters of
the failed waveform test. For a sampled stable filter, $z_n=qz_{n-1}+(1-q)x_n$
with $q=\exp(-\Delta t/\tau)$, it retained three functions of $\log(\tau)$: the zero-initial-
state terminal response B, an initial-state/residual cross-term H/N, and the
zero-state mean squared loss R/N. Deterministic quantities $A=q^N$ and
$P=\sum_{i=1}^{N}q^{2i}$ depend only on the model and sample count.
For incoming state s, terminal state is $As+B$ and mean loss is
$(P/N)s^2+2(H/N)s+R/N$. Thus an unspecified incoming state is handled by exact
affine/quadratic algebra, while an unspecified time constant is handled by
interpolation over the declared interval [0.002,0.2] seconds. The encoder knows
the interval and model family; it does not know the eventual random queries.
It has already observed the target values used in the loss.
Four representations shared two equal scalar budgets: ordinary linear tables,
natural cubic tables, cardinal cubic Hermite values plus analytic derivatives,
and Chebyshev-Lobatto interpolation. Hermite used half as many nodes so that
derivatives were not free extra stored data. The 36-record confirmation panel
used six seeds, two lengths and three target regimes, including model
misspecification. Every record was encoded directly and as 32 blocks merged
sequentially and in a balanced tree.
| Representation | Bytes | Passing versions / 108 | Worst mean-loss error |
|---|---:|---:|---:|
| Linear, 32 scalars/function | 802 | 0 | 3.64e-4 |
| Linear, 64 | 1,570 | 0 | 8.80e-5 |
| Natural cubic, 32 | 802 | 45 | 9.70e-5 |
| Natural cubic, 64 | 1,570 | 108 | 2.26e-5 |
| Cardinal Hermite, 32 | 802 | 108 | 7.88e-6 |
| Cardinal Hermite, 64 | 1,570 | 108 | 4.17e-7 |
| Chebyshev, 32 | 802 | 108 | 4.60e-11 |
| Chebyshev, 64 | 1,570 | 108 | 2.27e-14 |
Passing also required terminal error and optimizer regret at most 1e-4,
nonnegative quadratic form within tolerance, successful optimization and a
serialized object no larger than 2,048 bytes. Mean-loss error alone therefore
does not explain every failure in the table. Both Hermite budgets, both
Chebyshev budgets and the larger natural cubic table passed every capability
check and the computational gate. No materially indefinite quadratic was
observed at the tested queries.
For 259 warm queries at N=65,536, median paired speedups over the compiled
raw-data reference were approximately 634x for Hermite-32 and 218x for
Chebyshev-32. Whole-record encoding amortized after median 0.075 and 0.126
such query batches respectively. These are implementation-specific repeated-
query savings, not speedups over the best possible competing summary, and
not evidence that preprocessing is free for a single question. All block
encoding and merge costs remain in the complete records.
The comparison is scientifically useful precisely because it does not force
a spline win. Hermite offers fast local evaluation and a smaller-budget
advantage over an ordinary natural cubic table. Chebyshev exploits the analytic
response functions much more accurately. Future implementations may change
their timing order. A useful computational representation should select the
appropriate approximation family instead of treating basis identity as its
success criterion.
## 6. Composition and the history-length bound
At fixed nodes, evaluation preserves sums and pointwise products. Reconstructing
an interpolant and evaluating it again at those nodes returns the same values.
The block-merge formulas depend only on these values and deterministic metadata.
Consequently, interpolating after each merge gives the same nodal statistics
as computing the exact merged functions first and then interpolating once.
Uniqueness of the interpolant makes the final functions agree in real arithmetic.
Chronological order remains essential; only parenthesization is interchangeable.
Hermite data obeys the analogous first-jet algebra: the derivative of a product
uses the product rule. This provides the same composition property when value
and derivative nodes remain fixed. It is not a property of arbitrary oblique
or orthogonal projection. For example, $L^2$ projection onto $\mathrm{span}\{1,u\}$ on [-1,1]
gives $P(u^2)=1/3$, so $P(P(u^2)P(u))=u/3$, whereas $P(u^3)=3u/5$. That counterexample
disproves the required commutation identity for a generic projection.
For the stable single-pole family, analytic continuation supplies a useful
uniform bound. With bounded inputs and targets, a suitable complex ellipse
in log(tau) retains a uniformly stable filter. Its response, normalized cross-
term and normalized loss are bounded independently of the number of samples.
The classical Chebyshev analytic-function theorem then gives geometric decay
of interpolation error with degree.[^19] The full derivation, including a fixed
ellipse parameter and amplitude factors, is in `THEORY_03.md`.
The exposed-record audit found node-value and Hermite-derivative discrepancies
below 2.60e-14. All observed Chebyshev errors lay below the derived real-
arithmetic bounds. This is a post-exposure explanation, not another independent
confirmation panel. Floating-point node evaluations, transforms and merges
are not formally certified. The mean-loss bound does not give an N-independent
absolute total-loss or posterior bound. A fixed amplitude envelope and fixed
one-dimensional query family are essential assumptions.
## 7. Long-stream finite checkpoints
A separate numerical scope test encoded bounded observations in chunks of
1,024 samples, restoring the current summary from a quantized checkpoint before
each subsequent merge. It examined both interpolation families with float64
and float32 checkpoint payloads, while keeping computation in float64. Twelve
fresh records used six independent seeds, constant or switching target dynamics,
and prefixes from 1,024 to 1,048,576 samples. Queries were selected only after
all checkpoints were saved.
| Checkpoint payload | Bytes | Worst terminal error | Worst mean-loss error |
|---|---:|---:|---:|
| Chebyshev, float64 | 822 | 3.69e-10 | 9.37e-12 |
| Chebyshev, float32 | 438 | 2.03e-8 | 1.30e-8 |
| Hermite, float64 | 822 | 1.95e-5 | 4.42e-6 |
| Hermite, float32 | 438 | 1.96e-5 | 4.43e-6 |
All four arms passed all 48 prefix/record checks. The changed header includes
known amplitude envelopes and a larger length field, so the byte counts are
not a revision of the preceding experiment's format. Independent SciPy
statistics checks agreed with the compiled reference to within 1.45e-15.
The largest stream represents about 17.5 minutes at the stated sampling rate,
not an indefinite-duration test.
This closes a specific implementation gap: input working chunks and retained
summaries need not grow with the tested history. The validation phase still
regenerated raw observations to check the claims, and the full process used
up to 180.6 MiB. The first encoding phase's process high-water mark was 106.4
MiB, dominated by the research environment rather than the serialized object.
Neither number is an embedded-device power or isolated allocator measurement.
Preserving an old average objective under target drift also does not establish
a good adaptation policy. A learner may need recency weighting, change-point
reasoning or a revised loss, none of which can be recovered from missing
statistics by naming the object a memory. The result concerns faithful queries
inside the fixed contract, not unrestricted continual learning.
## 8. High-impact application assessment
The closest supported vision is revisable scientific telemetry: an instrument
can transmit a small evidence object rather than a long recording, and a
recipient can later reconsider a meaningful physical model. The value would
come from preserving scientific conclusions under a communication or memory
limit, not from displaying a large reduction ratio for a single statistic.
Noise-aware inference and uncertainty are important because many consequential
decisions depend on distinguishing competing dynamics rather than merely
minimizing a scalar loss accurately.
| Vision | Decisive capability | Mandatory comparison |
|---|---|---|
| Revisable telemetry | Later physical re-estimation with reliable uncertainty | State-space/semiseparable inference, fixed parameter banks and compressed raw evidence |
| Portable machine experience | Useful learning after episodes are exchanged | Fixed-feature Gram memory, recursive identification and local replay |
| Computational simulation archive | Later operators and derived quantities remain accurate | MGARD/ZFP, predictive codecs and reduced models |
| Compressed neural inference | Future sequence/model decisions survive smaller state | Attention/cache methods, quantization and distillation |
These are alternative destinations, not interchangeable names for the current
scalar mechanism. The failed waveform experiment does not admit a simulation-
archive claim. The passed objective experiment does not admit a neural-cache
claim. A small state on a laptop does not establish edge-device energy savings.
The role of a vision is to select consequential tests, not to convert every
local improvement into evidence for the entire vision.
A tempting next step would add many first-order filters after compression and
claim architectural growth. That is insufficient on its own. If the new filter
trajectories are interpolated from K retained node trajectories, they remain
in the same K-dimensional span. Their quadratic losses can be represented by
an ordinary Gram matrix and cross-vector. A free linear readout of the fixed
bank contains every such interpolated mixture and is the mandatory control.
Changing a pole's name or the mixture width does not create new evidence.
The more demanding question is whether a compact family of composable physical
inference objects retains enough flexibility for an application that existing
summaries cannot serve as well. Linear-Gaussian block factors are one plausible
bridge: fixed-parameter elimination is structured, but parameterized likelihoods
introduce normalization, conditioning and uncertainty requirements absent from
the current least-squares test. This is a prospective hypothesis, not an
implemented result or an established novelty claim.
In particular, observation-count independence on a fixed spatial domain must
not be confused with independence of a stream's increasing duration. A future
application must separate those axes. The current scalar mean-loss result
does not prove a duration-independent posterior representation. The additional
competitive assessment in `NEXT_COMPARATOR_AUDIT.md` therefore does not admit
a generic Gaussian-inference panel without a sharper capability requirement.
## 9. Next experiment and stopping criteria
The next admitted application should involve noisy physical time series,
meaningful competing parameter explanations and a hard retention or transmission
budget. Before numerical comparison, it needs a precise target inference,
timestamp and missingness semantics, parameter/noise domain and required
accuracy of the conclusion. Posterior or likelihood-ratio tasks must control
the relevant total error; a small per-sample discrepancy can become important
after multiplication by a long history.
The reference should exploit the same physical structure. For exponential
kernel families, this means state-space or semiseparable computation rather
than an unnecessarily dense solver. A plain parameter bank with the same
interpolation and byte accounting is also required. If it reproduces the
proposal, novelty must lie in a demonstrably different capability, guarantee
or resource tradeoff, not terminology. Ingestion, encoding, query costs and
any recoverable residual evidence all belong in the comparison.
A convincing outcome would allow the recipient to make several later-selected
scientific inferences at near-reference quality while using materially less
transmission, retention and repeated work than those controls. Episodes or
instruments would remain grouped in evaluation, with independent held-out
cases and all declared query families reported. A failure of uncertainty or
coverage would remain a failure even if a headline average error were small.
The present scalar screens are closed. No additional degree sweep, easier
target, optimizer search or dataset subset is justified by them. The evidence
supports a concrete direction and a reusable implementation, but the remaining
gap is application-level utility and novelty. A high-impact compression result
would emerge from crossing that gap, not from making an already accurate
one-parameter surrogate marginally smaller.
## 10. Evidence package and scope
The research package contains separate protocols, complete numerical records,
actual serialized objects, source hashes, analysis scripts and tests. Study 01
has 864 checked codec outputs; study 02 has 10,080 checked block/final memory
objects; the algebra audit is explicitly post-exposure; study 04 has 192 checked
long-stream checkpoint objects. Repeated queries, merge orders and related
target regimes are not treated as independent experimental subjects.
The numerical work uses one process/thread and the existing environment.
No reserved trading data, production systems or neural training checkpoints
are involved. The tested compression mechanisms are synthetic and deliberately
bounded. Results and failures are preserved without reopening earlier panels.
Reproduction instructions and artifact accounting are provided in `README.md`;
reading depth and prior-art limits are recorded separately in `READING_LEDGER.md`.
## Sources
[^1]: Pedram Pad and Michael Unser. [Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes](https://bigwww.epfl.ch/publications/pad1501.pdf). IEEE Transactions on Signal Processing, 2015. DOI 10.1109/TSP.2015.2447494.
[^2]: Pier Luigi Dragotti, Martin Vetterli and Thierry Blu. [Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang-Fix](https://bigwww.epfl.ch/publications/dragotti0701.pdf). IEEE Transactions on Signal Processing, 2007.
[^3]: Ildar Khalidov and Michael Unser. [From Differential Equations to the Construction of New Wavelet-Like Bases](https://bigwww.epfl.ch/preprints/khalidov0501p.pdf). Preprint associated with the 2006 publication. Mariantonia Cotronei and Nada Sissouno. [A note on Hermite multiwavelets with polynomial and exponential vanishing moments](https://arxiv.org/abs/1702.01007), 2017.
[^4]: Michael Unser. [Quasi-Orthogonality and Quasi-Projections](https://bigwww.epfl.ch/publications/unser9608.html). Applied and Computational Harmonic Analysis 3, 201-214, 1996. Author abstract on quasi-projection and short quasi-duals.
[^5]: Emmanuel Candes, Yonina Eldar, Deanna Needell and Paige Randall. [Compressed sensing with coherent and redundant dictionaries](https://arxiv.org/abs/1005.2613). 2010 preprint.
[^6]: Remi Gribonval, Gilles Blanchard, Nicolas Keriven and Yann Traonmilin. [Compressive Statistical Learning with Random Feature Moments](https://arxiv.org/abs/1706.07180). 2017 preprint and subsequent revisions.
[^7]: Jonathan H. Huggins, Ryan P. Adams and Tamara Broderick. [PASS-GLM: polynomial approximate sufficient statistics for scalable Bayesian GLM inference](https://www.cs.princeton.edu/~rpa/pubs/huggins2017pass.pdf). NeurIPS, 2017.
[^8]: Yusuf Bugra Erol. [Joint State and Parameter Estimation in Temporal Models](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2018/EECS-2018-28.pdf). UC Berkeley technical report, 2018; chapter 3, particularly sections 3.3-3.5.
[^9]: Lawrence Livermore National Laboratory. [ZFP project](https://computing.llnl.gov/projects/zfp). Compressed floating-point arrays and access/computation documentation; accessed September 2026.
[^10]: MGARD authors. [MGARD: A multigrid framework for high-performance, error-controlled data compression and refactoring](https://arxiv.org/abs/2401.05994), 2024. Xuan Wu et al. [Error-controlled Progressive Retrieval of Scientific Data under Derivable Quantities of Interest](https://arxiv.org/abs/2411.05333), SC 2024.
[^11]: Fabio Nobile, Raul Tempone and Clayton G. Webster. [A sparse grid stochastic collocation method for elliptic partial differential equations with random input data](https://www.math.fsu.edu/~aluffi/archive/paper290.old.pdf). Author preprint dated June 22, 2006; related journal publication 2008.
[^12]: [Temporal Parallelisation of the HJB Equation and Continuous-Time Linear Quadratic Control](https://arxiv.org/abs/2212.11744), 2022 preprint. Tri Dao and Albert Gu. [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://openreview.net/pdf?id=ztn8FCR1td), 2024.
[^13]: Daniel Foreman-Mackey et al. [Fast and scalable Gaussian process modeling with applications to astronomical time series](https://arxiv.org/abs/1703.09710). 2017. [Celerite kernel documentation](https://celerite.readthedocs.io/en/stable/python/kernel/) supplies the exponential/oscillatory kernel interface.
[^14]: Philip Greengard. [Efficient Fourier representations of families of Gaussian processes](https://arxiv.org/pdf/2109.14081). 2021 preprint, inspected revision dated May 31, 2024; sections 2-3 and Algorithm 1.
[^15]: Kun Dong et al. [Scalable Log Determinants for Gaussian Process Kernel Learning](https://proceedings.neurips.cc/paper_files/paper/2017/file/976abf49974d4686f87192efa0513ae0-Paper.pdf). NeurIPS, 2017; sections 3.2 and 3.5. Simo Sarkka and Angel F. Garcia-Fernandez. [Temporal Parallelization of Bayesian Smoothers](https://arxiv.org/abs/1905.13002). 2019 preprint.
[^16]: Krzysztof Choromanski et al. [Rethinking Attention with Performers](https://arxiv.org/abs/2009.14794). 2020 preprint; ICLR 2021.
[^17]: [KVzip: Query-Agnostic KV Cache Compression with Context Reconstruction](https://arxiv.org/abs/2505.23416). 2025 preprint. Screened as competitive context, not benchmarked here.
[^18]: Themistoklis Haris and Krzysztof Onak. [Compression Barriers for Autoregressive Transformers](https://arxiv.org/abs/2502.15955). 2025 preprint. Its stated lower bounds depend on the attention/query and dimension assumptions.
[^19]: Lloyd N. Trefethen. [Approximation Theory and Approximation Practice, chapter 8 source](https://raw.githubusercontent.com/chebfun/ATAP/development/chap8.m). Theorem 8.2, equation (8.3), Chebyshev interpolation of analytic functions.