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TOPIQ: Statistical Error Propagation for Quantity-of-Interest Prediction under Lossy Compression

Youyuan Liu, Bo Jiang, Taolue Yang, Sheng Di, Robert Underwood, Sian Jin

arXiv:2608.26912v1cs.PFcs.ITmath.NA

TL;DR

Per-element compression bounds do not determine the reliability of downstream QoIs, especially for dynamic queries and derived analyses. TOPIQ propagates compact error statistics through runtime-composed primitive-operator graphs, producing QoI-level bias and uncertainty. Across broad evaluations, it achieves calibrated predictions for most configurations and supports fast post-hoc confidence analysis, with limitations when error statistics are strongly spatially non-stationary.

  • Problem

    Per-element error bounds and traditional quality metrics do not directly characterize compression effects on downstream QoIs such as aggregates, neural-network predictions, and multi-field quantities.

  • Method

    TOPIQ decomposes runtime-defined QoIs into primitive operators and propagates a compact error tuple using closed-form rules that model spatial correlation and data–error coupling.

  • Results

    93.1% of 552 evaluations achieve well-calibrated predictions, while pre-computed metadata enables post-hoc uncertainty quantification at 56x–402x speedup over direct computation.

  • Takeaways & Limitations

    Runtime composition and compact metadata allow TOPIQ to provide uncertainty estimates for arbitrary post-hoc query regions without retaining the original data.

  • Takeaways & Limitations

    A single global correlation index can misrepresent strongly spatially non-stationary errors, producing under-calibrated predictions in heterogeneous fields.

Abstract

from arXiv · show

Lossy compression is essential for managing massive scientific data, but per-element error bounds do not translate into bounds on downstream quantities of interest (QoIs) such as regional averages, neural network predictions, or multi-field derived quantities. We present TOPIQ, a statistical error-propagation framework that predicts QoI-level bias and uncertainty from compact compression metadata (less than 0.1% of original data). TOPIQ decomposes QoIs into primitive operators with closed-form propagation rules accounting for spatial error correlation and data-error coupling; new QoIs are supported by composition at runtime with no per-QoI derivation or retraining. Across 552 evaluations spanning 4 datasets, 3 compressors, 4 QoI families, and 8 error bounds, 93.1% of configurations achieve well-calibrated predictions. Pre-computed metadata enables post-hoc uncertainty quantification for arbitrary query regions at 56x-402x speedup over direct computation. A case study demonstrates integration into an AI-driven analysis pipeline with end-to-end confidence intervals for dynamically composed queries.

I. INTRODUCTION

Scientific simulations generate data volumes that make lossy compression necessary, but data-level error bounds do not guarantee reliability for downstream QoIs. TOPIQ addresses this gap by predicting QoI-level bias and uncertainty for dynamically defined analyses from compact metadata.

  • Motivation: Climate, cosmology, and fusion simulations generate data at scales that create fundamental storage, transfer, and post-hoc analysis challenges.Reported examples range from hundreds of terabytes every 16 seconds to petabyte- and exabyte-scale campaigns.
  • Motivation: Data-level error bounds do not directly guarantee reliability for regional averages, derived physical quantities, or inference models that drive scientific interpretation.Traditional field-level metrics cannot characterize downstream QoI effects.
  • Dynamic analysis: Dynamic workflows query arbitrary locations and define QoIs at runtime, making it impractical to fix every analysis target before compression.This need is especially relevant to AI-driven and autonomous analysis pipelines.
  • Motivation: QoI distortion depends on computational structure, spatial error correlation, and coupling between data values and compression errors.Spatial aggregation can amplify pixel-level errors by orders of magnitude when errors are correlated.
  • TOPIQ: TOPIQ predicts calibrated confidence intervals for runtime-defined QoIs expressible through supported smooth primitive operators using metadata smaller than 0.1% of the original data.Its operators use closed-form propagation rules accounting for spatial correlation and data–error coupling, while new QoIs are composed without per-QoI derivation or retraining.
  • QoI scope: QoIs span pointwise transforms, regional aggregates, multivariate derived quantities, and complex compositions such as climate-analysis ratios.This spectrum motivates propagation methods that operate beyond single-field pointwise metrics.

C. Predicting Compression Impact in Scientific Workflows

Existing quality metrics and QoI-preserving compressors do not cover dynamically composed, post-hoc queries without fixing the QoI and region during compression. TOPIQ instead propagates compact error statistics through computation graphs to estimate QoI-level bias and uncertainty.

  • Existing approaches: PSNR-oriented quality methods and compression-ratio predictors do not characterize downstream QoI effects such as error amplification through spatial aggregation.Their targets are predefined data-quality metrics rather than arbitrary downstream computations.
  • QoI-preserving compression: QoI-preserving compression derives sufficient pointwise error bounds from a specified QoI and uses them to guide per-pixel compression.This approach provides deterministic QoI guarantees in addition to data-level error bounds.
  • TOPIQ workflow: TOPIQ separates one-time offline metadata extraction from online query-time uncertainty analysis for post-hoc QoIs.The online stage selects metadata granularity, builds an error-propagation graph from the QoI, and produces a confidence interval.
  • Limitations of prior methods: QoI-preserving methods require the QoI and analysis region to be fixed at compression time, and each new QoI requires case-specific pointwise-bound derivation.They remain appropriate when the target is known in advance but are constrained for dynamic exploratory workflows.
  • Error representation: TOPIQ uses a five-tuple T = (µ, σ2, b, v, c) as the universal interface passed through computation-graph operators.The signal pair (µ, σ2) is separated from error and data-coupling statistics (b, v, c).
  • Error representation: TOPIQ propagates measured global bias because some data/compressor combinations produce genuine systematic compression bias.The bias is retained alongside spatially varying block-level fluctuations.

B. Single-Field Operators

Single-field operators propagate compression-error statistics through a shared tuple, using exact rules for linear functions and second-order approximations for nonlinear ones. Shared errors in repeated references to one field require distinct treatment because independent-copy models underestimate variance.

  • Tuple interface: The tuple T = (µ, σ2, b, v, c) is the universal input and output interface for operators.Each operator returns the same five-component structure for composition through the computation graph.
  • Propagation rules: Linear operators propagate exactly, while nonlinear operators such as x2, σ(x), and 1/x use a second-order Taylor expansion.Constant addition shifts only the mean, and scalar multiplication scales tuple components by the appropriate powers of its factor.
  • Nonlinear bias: Second-order bias combines propagated input bias, a variance correction, and a data-error coupling correction.The terms are f′ · b, 1/2f′′ · v, and f′′ · c, respectively.
  • Shared-error effects: Shared compression errors in x2 make its output variance 4(µ2 + σ2)v, twice the independent-copy result 2(µ2+σ2)v.The cross term cannot be dropped because both factors reference the same compressed field and therefore share identical errors.

C. Multi-Field Operators and Cross-Field Independence

Multi-field operators treat errors from independently compressed fields as cross-field independent, while retaining within-field data-error covariance. This assumption preserves composability and extends the primitive operator framework to general multi-field QoIs.

  • Cross-field independence: Independent compression runs motivate treating each field’s error as independent of the other field and its data-error pair.Original fields may be physically correlated, but their independently generated compression errors exhibit substantially weaker cross-field correlation.
  • Combination rules: Under cross-field independence, linear and nonlinear cross-field operations use the same combination rules.The key distinction is independence between error sources rather than the operator’s linearity.
  • Compositional support: TOPIQ composes supported primitives to handle general multi-field QoIs such as the cloudy ratio.This extends beyond the limited multi-field support described for existing QoI-preserving compressors.
  • Addition and subtraction: For x ± w, the two fields’ error variances add because their errors are uncorrelated under the independence assumption.The supplied passage introduces the operator and its independently compressed inputs; the variance rule is detailed in the following addition/subtraction passage.

1) Addition and Subtraction:

Addition and subtraction combine independently compressed fields through cross-field error terms, with variance contributions from the two fields adding regardless of the sign. Multiplication retains within-field effects but drops cross-field covariance terms under independence, while division composes reciprocal and multiplication.

  • Addition and subtraction: The variances σ2 and v add for x ± w regardless of whether the operator is addition or subtraction.This follows because the two fields’ errors are uncorrelated under the cross-field independence assumption.
  • Multiplication: Multiplication expands the output error into leading, within-field, and cross-field covariance groups.The within-field group uses carried quantities, whereas the cross-field group requires covariances that are dropped under independence.

3) Division:

Aggregation requires modeling spatially correlated errors within a field, because independent-error assumptions can severely underestimate block variance. TOPIQ separates uncorrelated and common-mode components, calibrates their ratio once, and propagates both through pixel-wise operations before aggregation.

  • Why aggregation is special: Spatially correlated errors can make aggregated variance exceed the independent-error estimate n·v by orders of magnitude.Errors within one compression run share structure across pixels, unlike errors from independent cross-field compression runs.
  • Error decomposition: Exact aggregation variance requires O(n2) pairwise covariances, so TOPIQ uses a parsimonious vu/vc decomposition.vu averages away like i.i.d. noise, while vc persists like a common-mode offset.
  • Calibration: The correlation ratio r = vc/v = (α −1)/(n −1) is approximately scale-invariant and transfers to new block sizes without re-measurement.TOPIQ estimates α during offline metadata extraction and reuses it for arbitrary runtime query blocks.
  • Composition: Pixel-wise operators preserve the vu/vc correlation decomposition before aggregation, allowing their outputs to feed subsequent operators.The evaluation uses block sizes of 200×200 for 2D and 323 for 3D, where the model is reported as well-calibrated.
  • Aggregation propagation: For block sums, vu scales linearly and vc quadratically; for block means, the variance is v̄δ = vu/n + vc.This captures behavior from independent errors through perfectly correlated errors.

E. Composing QoIs from Primitive Operators

TOPIQ represents supported QoIs as directed acyclic graphs of smooth primitive operators and propagates bias and variance by composing their closed-form rules. This supports runtime-defined single-field, neural-network, and multi-field QoIs without per-QoI derivation or retraining.

  • Any QoI expressible as a directed acyclic graph of supported smooth operators can be handled by composing propagation rules.Each node transforms input tuples into an output tuple, and the final node yields QoI-level bias and variance.
  • Longer multi-field QoIs, including the six-operator cloudy ratio, achieve calibration comparable to single-operator QoIs.The retained first- and second-order terms propagate bias and variance across multi-step pipelines.
  • TOPIQ composes weighted-sum, sigmoid, and linear operators for neural-network QoIs and subtraction, masking, multiplication, summation, and division for cloudy ratios.These examples extend beyond native support in existing QoI-preserving compressors.
  • Figure 3 shows each operator transforming input tuple(s) into output tuples, enabling automatic composition without QoI-specific derivation.

A. Setup and Metrics

The evaluation spans four scientific datasets and three compressors using standardized residuals, bias, calibration, and coverage metrics. The ablations show that spatial error correlation and data–error covariance are necessary for reliable QoI uncertainty estimates, especially when correlations are large.

  • A. Setup and Metrics: Four scientific datasets and three compressors are evaluated across relative error bounds from 5×10−3 to 10−5.The datasets include CESM-ATM, NYX, SCALE-LETKF, and Hurricane-ISABEL; compressors are SZ3, SPERR, and ZFP.
  • A. Setup and Metrics: Standardized residuals use z = (q∗−ˆq)/ˆσ, with ideal bias µz = 0, calibration metric σz = 1, and 3σ coverage of 99.7%.
  • B. Ablation Study: Ignoring spatial correlation can inflate σz from 1.0 to 14.7 when ZFP/CLDTOT has α = 218, while small α produces minimal impact.Random permutation collapses α to approximately 1, confirming the correlation is spatial.
  • B. Ablation Study: Ignoring data–error covariance shifts µz to −8.72, whereas ignoring α inflates σz to 7.9 for NYX dark-matter density compressed with SPERR.TOPIQ corrects both effects, and permuting errors restores σz ≈1.
  • B. Ablation Study: At εrel ≲ 10−4, both ablation errors recover as SPERR’s α drops from 53 to 2 and |Cov(x, e)| decreases by four orders of magnitude.The recovery trend is compressor-specific.

C. Prediction Accuracy

TOPIQ evaluates increasingly complex QoIs built from primitive operators, including aggregation, neural networks, cross-field weighted sums, and cloudy ratios. Its calibration remains strong across configurations and error bounds, with 94.2% of εrel=10−3 configurations within the stated σz range.

  • C. Prediction Accuracy: TOPIQ evaluates four QoI families, including mean-square, a 1.28M-parameter one-hidden-layer MLP, cross-field weighted sums, and cloudy ratios.The neural network uses weighted-sum aggregation, while the cloudy ratio combines smooth operations across three fields.
  • C. Prediction Accuracy: TOPIQ keeps σz within [0.8, 1.2] from εrel=5×10−3 to 10−5 for twelve mean(x2) dataset/compressor combinations.Other QoI families exhibit similar stability.
  • C. Prediction Accuracy: The cloudy ratio uses the operator chain sub → mul → sigmoid → mul → sum → div across FLUT, FLUTC, and CLDTOT.
  • C. Prediction Accuracy: 94.2% of configurations at εrel=10−3 achieve σz ∈[0.7, 1.3], with median observed 3σ coverage of 99.6%.Some compressors introduce systematic bias, which TOPIQ propagates unconditionally.

1) Calibration Across Error Bounds:

TOPIQ remains well calibrated across wide error-bound ranges and supports arbitrary spatial queries through interpolated metadata. Its main failures arise from spatially non-stationary errors and neglected cross-field correlations.

  • Calibration results: 93.1% of 552 evaluations fall within σz ∈[0.7, 1.3] across eight error bounds.At εrel=10−3, three of 69 configurations are under-calibrated and one is over-conservative.
  • Failure modes: SPERR on SCALE-LETKF PRES is systematically under-calibrated because one global α cannot represent strongly heterogeneous local error structure.Per-block standard deviation varies 939× and per-block α varies up to 9×, producing σz ≈1.8.
  • Practical guidance: A high coefficient of variation in per-block error variance flags spatial non-stationarity; switching to a more uniform compressor is recommended.The guidance targets the SPERR failure mode observed on SCALE-LETKF PRES.
  • Failure modes: Cross-field independence can yield conservative or optimistic uncertainty for multi-field ratios when field-error correlations are omitted.For ZFP’s cloudy ratio on CESM-ATM, systematic field biases partially cancel and TOPIQ predicts σz ≈0.50.
  • Arbitrary spatial queries: Median |∆σz| remains below 0.04 for arbitrary 200×200 queries using interpolated metadata.The metadata grid is 150×150 and remains below 0.1% of original CESM field data.
  • Arbitrary spatial queries: 3σ coverage remains ≥99% across compressors and error bounds for meta-interpolated predictions.This extends uncertainty prediction beyond block-aligned queries to regions unknown at compression time.

4) Prediction Overhead:

TOPIQ makes post-compression uncertainty prediction practical by using compact metadata rather than the original and decompressed arrays. It also supports dynamically composed queries, including LLM-driven pipelines, with calibrated confidence intervals.

  • Prediction overhead: 56×–402× speedup over direct QoI error computation is achieved for 200×200 CESM-ATM blocks.Prediction costs O(1) floating-point operations per query, compared with O(n) for direct computation.
  • Prediction overhead: < 0.1% of original data is required as pre-computed metadata for post-hoc uncertainty quantification.Metadata pre-computation takes approximately 33 ms per field; direct computation becomes impossible after the original data is discarded.
  • AI-driven analysis: TOPIQ maps each computation step in an LLM-decomposed climate query to a primitive operator.The example produces a 99.7% confidence interval from compression metadata alone.
  • AI-driven analysis: TOPIQ supports uncertainty quantification for dynamically constructed QoIs without re-compression or QoI-specific derivation.This contrasts with QPET, which requires the QoI and analysis region to be fixed at compression time.
  • Comparison with QoI-preserving compression: 99.9% 3σ coverage is achieved on random off-grid queries with SZ3 and TOPIQ meta-interpolation.For CLDTOT at εrel=10−3, σz = 0.97, whereas QPET leaves up to 11% of queries exceeding its grid-level worst case.
  • Future scope: Current second-order Taylor propagation targets smooth functions and excludes operators such as ReLU, max, min, and discrete feature counts.The paper identifies piecewise local rules or richer distribution summaries as possible future approaches.
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