Research & AlgorithmsDANIEL SCHMITTER · RESEARCH & ENGINEERING

Numerical intelligence · Research & Algorithms

Return an error bound with the prediction

For a portfolio, the useful error bound is often the bound on the combined response—not the sum of pessimistic bounds on every leg.

Daniel Schmitter · Research story · Revised draft

Read the paper ↗Paper PDF
EXPLORE THE IDEA

Preserve the cancellation

The quantity you ask for has its own error geometry.

SIGNED READOUTTwo nearby requests can cancelCopper minus blue: the combined questionCOMBINE BEFORE TAKING A NORMDo not discard useful cancellationSigned L² norm 0.4676Illustrative L² geometry, not the certified bound
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Computed signed Gaussian readout example: two unit-height bumps approach and cancel. The displayed signed L² norm is obtained by numerical integration of the illustrated functions; it is not the paper’s certified Green-kernel bound.

Follow the information

From input to outcome

Cancellation is retained inside each quadratic form before combining error allowances. This bounds the requested signed functional rather than summing separately pessimistic leg bounds. It controls numerical error under the supplied model, not model validity.

Scroll the diagram horizontally to follow the route. Keyboard: focus the diagram, then use the arrow keys.

Signed source + readout → Green-kernel forms → Spectral error enclosure → Complete allowance → Value with error interval. Cancellation is retained inside each quadratic form before combining error allowances. This bounds the requested signed functional rather than summing separately pessimistic leg bounds. It controls numerical error under the supplied model, not model validity.
Information-flow map. Tighter bounds are not the same as faster evaluation. Original vector schematic based on the method and evidence discussed in this article; signal shapes and icons are illustrative, not additional measurements. Open full-size diagram ↗

Read the main route from left to right; labelled side branches show additional inputs, checks or feedback. The sections below explain the operations and their experimental limits.

Precision has a shape

Suppose two option positions largely offset each other. Bounding each position’s numerical error separately and then adding the bounds is safe, but it may discard exactly the structure that makes the portfolio small. The same issue appears when subtracting nearby predictions, computing a sensitivity, or comparing two continuous models. The output is a signed functional, not a bag of unrelated values.

Ask for the portfolio response directly

A spread is a signed combination of payoffs. A finite spot response is another signed combination of readouts. Bounding each atom first throws away cancellation. The operator energy norm instead evaluates a small Green-kernel quadratic form on each complete signed functional, then combines those norms with a scalar time-inversion error bound.

The resulting allowance follows the requested output rather than the largest individual component. Its tightness gain is not the same as a latency gain. Boundary assumptions and outward arithmetic still have to be included before the value becomes a usable numerical enclosure.

Search freely; verify a small assertion
Search freely; verify a small assertion. Original scientific diagram; the stated component and information flow, not an additional experiment. Open full-size figure ↗

Use the geometry of the operator

For the declared diffusion, a spectral argument transfers a scalar time-approximation error to an output error through two energy norms: one for the source and one for the readout. Each norm is a signed quadratic form in a Green kernel. Opposite-sign terms are combined inside that form, before any triangle inequality throws their cancellation away.

∣response error∣≤δ ∥signed source∥K−1 ∥signed readout∥K−1\begin{gathered}|\text{response error}|\le\delta\,\|\text{signed source}\|_{K^{-1}}\,\|\text{signed readout}\|_{K^{-1}}\end{gathered}
A uniform scalar spectral error becomes an output-specific bound. Signed Green inner products preserve cancellation before norms are combined.

A bound over the spectrum, not a sampled check

The implementation encloses the scalar approximation error over the entire nonnegative spectrum using interval Taylor expansions and a tail treatment. It then adds validated solve, arithmetic, and any applicable domain allowances. Agreement on a handful of eigenvalues would be a diagnostic, not the same guarantee. Exact spatial cells remove one approximation source but do not make floating-point computation exact.

What improved

In the initial signed-output example, this transfer is about 172 times tighter than separately accumulating leg bounds. That is a tighter guarantee, not a 172-times runtime speedup. A later fresh panel contains the independent reference in all 320 field/maturity cases. Its assumptions still include a bounded positive coefficient field, declared boundaries, and the specified arithmetic behavior.

Verified material directions in the single- and four-expiry synthetic studies. Missing witnesses remain unresolved; these counts are not upper bounds on model ambiguity.
Verified material directions in the single- and four-expiry synthetic studies. Missing witnesses remain unresolved; these counts are not upper bounds on model ambiguity. Open full-size figure ↗

An error budget shaped like the requested output

For scientific ML and numerical services, the broader idea is output-aware assurance. A precise bound on the quantity a user asks for can be far more useful than a crude uniform bound on every intermediate field. This work implements that principle for a supplied one-dimensional pricing operator, not for arbitrary neural outputs.

A useful interface for numerical intelligence

A learned system can propose a physical model or request a response; a numerical component can return both the value and the uncertainty caused by its computation. This does not quantify every modeling error, and it does not certify that the world obeys the supplied equation. Its value is narrower and tangible: subsequent decisions need not confuse a precise-looking decimal with a numerically justified one.

Evidence & further reading

The links below distinguish the project record from foundational literature. This revised story does not add a new application-validation experiment.

  1. Compact, verifiable operator-based options risk. Spline research archive (2026). Local archive snapshot.
  2. Compact operator representations for option pricing and risk. Spline research archive (2026). Local archive snapshot.
  3. Single-thread deployment succeeds on the shared workstation. Spline research archive (2026). Local archive snapshot.

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