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Joint Precision Neural Networks: Task-Aware Dependency and Predictive Learning

Andrea Cavallo, Samuel Rey, Antonio G. Marques, Elvin Isufi

arXiv:2610.06023v1cs.LG

TL;DR

PCA and regularized precision estimation have limitations for downstream learning: PCA is sensitive to finite-sample errors, while regularized precision estimates remain task-agnostic. The paper introduces PNNs and jointly optimizes precision matrices and network weights; PNN-Joint reports the lowest synthetic-task NMSE across tested conditions and the most consistent performance across real datasets.

  • Problem

    PCA is sensitive to finite-sample estimation errors, while regularized precision estimates do not incorporate the downstream objective and may be suboptimal for the learning task.

    Source: §I · p. 1, §IV · p. 4

  • Method

    PNN-Joint alternates between task-aware precision-matrix estimation and network-weight updates, using a graphical-lasso objective within a precision-based GNN framework.

    Source: A. Model Definition · p. 3, §IV · p. 4

  • Results

    PNN-Joint achieves the lowest downstream NMSE across synthetic perturbation levels and sample sizes and delivers the most consistent performance across real datasets.

    Source: A. Synthetic Data: Regression Task (O1) · p. 7, B. Real Data (O2) · p. 8

  • Takeaways & Limitations

    Task-aware optimization also improves recovery of the underlying graph structure, with better Frobenius errors and F-scores than baselines across the reported synthetic settings.

    Source: A. Synthetic Data: Regression Task (O1) · p. 7

Abstract

from arXiv · show

Exploiting meaningful latent structures from data to solve downstream tasks is a fundamental challenge in signal processing and machine learning. While Principal Component Analysis (PCA) and coVariance Neural Networks (VNNs) successfully leverage the covariance matrix to process data, they inherently capture both direct and indirect correlations. The precision matrix (inverse covariance) overcomes this by explicitly encoding conditional independencies, making it largely studied in graphical lasso and graph topology identification. However, finite-sample precision estimates are notoriously unstable, and regularized estimators remain task-agnostic. In this work, our principal contribution is tackling the challenging problem of task-aware graph inference. We propose Precision Neural Networks-Joint (PNN-Joint), a framework that jointly estimates a sparse, statistically grounded precision matrix alongside graph neural network weights via an alternating optimization scheme. As a foundational framework to support this, we introduce Precision Neural Networks (PNNs), a broader class of graph convolutional networks operating on precision estimators, and establish their spectral connections to PCA and VNNs alongside their stability to finite-sample errors. Extensive empirical evaluations on synthetic data, as well as real-world neuroimaging and motion sensor datasets, demonstrate that PNN-Joint yields highly interpretable task-aware graphs, exhibits remarkable robustness in low-data regimes, and consistently achieves the best or second-best performance among competitors on real-world tasks.

I. INTRODUCTION

The section motivates using statistical structure for downstream tasks and introduces PNNs and PNN-Joint to address limitations of covariance-based processing and task-agnostic precision estimation. It also reviews PCA and covariance filters as context for the precision-based approach.

A. Covariance Matrix and PCA

  • PCA is a rigid, task-agnostic operation whose spectral decomposition is sensitive to finite-sample errors, especially with small eigengaps or when dimensionality approaches sample size.PCA can also discard structurally relevant information when high-variance noise is present.Source: §I · p. 1, A. Covariance Matrix and PCA · p. 2

B. (Sparse) Covariance Filters and Neural Networks

  • VNNs learn end-to-end task representations and improve finite-sample stability, but covariance captures direct and indirect correlations, often producing dense graphs with spurious connections.This motivates moving from covariance-based processing toward the precision matrix.Source: §I · p. 1, B. (Sparse) Covariance Filters and… · p. 3

I. INTRODUCTION

  • PNN-Joint alternates sparse graphical-lasso-regularized precision estimation with network-weight updates, while PNNs use precision as the graph shift and connect spectrally to PCA and VNNs.The paper also reports finite-sample stability bounds and prediction quality among the top two models.Source: §I · p. 2

B. (Sparse) Covariance Filters and Neural Networks

  • Covariance filters learn polynomial responses over the covariance spectrum and can reproduce PCA filtering at sufficiently high order.The figure illustrates order-10 polynomials approximating PCA filtering.Source: B. (Sparse) Covariance Filters and… · p. 2, Figure 1 · p. 3
  • Bounding a covariance filter’s spectral Lipschitz constant makes output discrepancies scale with O(1/T), removing PCA’s restrictive eigengap dependency.Sparse covariance filters further reduce spurious correlations and computational overhead while retaining finite-sample stability guarantees.Source: B. (Sparse) Covariance Filters and… · p. 2

C. Precision Matrix and Graphical Lasso

The precision matrix offers a sparse representation of conditional dependencies, and graphical lasso estimates it with regularization. PNNs use precision-based graph convolutions to produce representations for downstream prediction.

C. Precision Matrix and Graphical Lasso

  • Under Gaussian assumptions, precision-matrix zeros indicate conditional independence, while graphical lasso promotes sparse estimates when finite-sample precision estimates are noisy.Its off-diagonal ℓ1 penalty controls structural sparsity and supports invertibility in high-dimensional regimes.Source: C. Precision Matrix and Graphical Lasso · p. 3

A. Model Definition

  • PNNs replace covariance with precision in graph convolution, using the conditional-dependence graph to propagate information between directly dependent variables.Sparse precision structure can enable efficient sparse matrix-vector multiplications.Source: A. Model Definition · p. 3
  • A PNN’s final representation is aggregated or passed through a readout layer to produce a downstream prediction.The resulting end-to-end map is denoted Φ(X, Θ, w), with trainable parameters w.Source: A. Model Definition · p. 3

B. Spectral Analysis and Connection to PCA and VNNs

PNNs use the covariance eigenbasis with inverse eigenvalues, allowing learned precision filters to modulate PCA projections; at sufficiently large filter orders, PNNs and VNNs can represent the same spectral transformations. PNN-Joint makes precision estimation task-aware through a relaxed alternating optimization, trading additional training time and tuning for a precision estimate that can differ from the graphical-lasso solution.

B. Spectral Analysis and Connection to PCA and VNNs

  • PNNs share PCA’s eigenbasis but use inverse covariance eigenvalues, and their learned filters can flexibly modulate PCA projections.At sufficiently large filter orders, PNNs and VNNs can represent the same spectral transformations, though their inductive biases may yield different downstream performance.Source: B. Spectral Analysis and Connection to… · pp. 3–4

IV. OPTIMIZATION ALGORITHM

  • PNN-Joint jointly estimates precision and neural weights so the precision topology is optimized for the downstream task.Its objective combines task loss and graphical-lasso loss, with α balancing task-driven and model-driven terms.Source: §IV · p. 4

A. Relaxed Formulation and Optimization

  • Relaxation decouples task loss from graphical-lasso loss with a quadratic penalty, enabling alternating updates of precision, an auxiliary matrix, and network weights.The precision update uses a graphical-lasso-related convex problem; the network-weight update is standard neural-network training.Source: A. Relaxed Formulation and Optimization · pp. 4–5
  • The alternating scheme can accommodate a task precision that differs from the graphical-lasso estimate, but adds a tuning parameter and increases training time.The iteration-dependent penalty is smaller early for exploration and larger later to bring the two matrices closer.Source: A. Relaxed Formulation and Optimization · p. 5

V. THEORETICAL ANALYSIS

The theoretical analysis studies how finite-sample precision errors affect precision-filter outputs and extends the bounds to full PNNs. It uses Lipschitz filters and assumes Gaussian observations with bounded covariance eigenvalues and a sparse true precision matrix.

V. THEORETICAL ANALYSIS

  • The analysis bounds how finite-sample precision-estimation errors propagate through precision filters and extends the results to full PNN architectures.It considers sample-inverse and graphical-lasso estimates, then the joint setting where precision and filter weights are estimated together.Source: §V · p. 5
  • A precision filter is Lipschitz when its frequency response changes by at most P times the change in eigenvalue over the specified interval.A larger P permits sharper frequency-response variation but increases sensitivity to eigenvalue perturbations.Source: §V · p. 5
  • The bounds assume Gaussian observations, bounded positive covariance eigenvalues, and a true precision matrix with S nonzero off-diagonal entries.The analysis also notes that precision estimates can perturb downstream performance.Source: §V · p. 5

A. Stability of Task-Agnostic PFs

For task-agnostic precision filters, the analysis compares sample-inverse and graphical-lasso estimates, with the latter providing a more favorable dimension dependence. For joint estimation, the stability bound also depends on the discrepancy between the task-aware and graphical-lasso-like precision estimates.

A. Stability of Task-Agnostic PFs

  • The sample-precision filter’s perturbation bound grows proportionally with graph size N and improves with sample count as 1/T.The text identifies high-dimensional regimes as challenging; for dense precision matrices, the sparsity advantage of the graphical-lasso bound vanishes.Source: A. Stability of Task-Agnostic PFs · p. 6
  • The graphical-lasso precision filter has a more favorable dependence on dimension N than the sample-precision filter because its precision estimate is sharper.When the true precision is sparse, its estimation error scales approximately as stated in the passage.Source: A. Stability of Task-Agnostic PFs · p. 6

B. Stability of PF-Joint

  • For joint precision estimation, the finite-sample error decreases at rate O(T^-1/2), while the bound retains a task-deviation term γ^(1−α)ξ.This result assumes the iterate matrices are within ξ in Frobenius norm; ξ measures the discrepancy between the task-aware and graphical-lasso-like estimates.Source: B. Stability of PF-Joint · p. 6
  • PF-Joint’s stability has a rate analogous to graphical lasso, with the additional discrepancy ξ; the bound depends on γξ, which can be controlled through γ.When the task requires a matrix close to the true precision, ξ is small and does not significantly affect the bound.Source: B. Stability of PF-Joint · p. 6

C. Stability of PNNs

PNN stability depends on precision-estimation error, with instability amplified by network depth and feature width. In synthetic regression, PNN-Joint performs best across the tested perturbation levels and sample sizes, while also improving precision recovery over the baselines.

C. Stability of PNNs

  • PNN filter instabilities are amplified by network depth L and feature size F, reflecting an expressivity-stability tradeoff in GNNs.The stability result applies under the bounded-filter and Lipschitz-nonlinearity assumptions stated for Proposition 3.Source: C. Stability of PNNs · p. 6

VI. NUMERICAL RESULTS

A. Synthetic Data: Regression Task (O1)

  • PNN-Joint achieves the lowest downstream NMSE across perturbation levels and sample sizes, with better precision-recovery metrics than the baselines.The recovery comparison covers Frobenius error and F-scores for both Θ and ˜Θ.Source: A. Synthetic Data: Regression Task (O1) · p. 7
  • PNN-Sample has the highest regression errors and weak precision recovery without many samples, while PNN-GL is more stable but task-agnostic.PNN-Sample and PNN-GL also show larger run-to-run standard deviations than PNN-Joint.Source: A. Synthetic Data: Regression Task (O1) · pp. 7–8

B. Real Data (O2)

Across brain-imaging and wearable-sensor tasks, PNN-Joint delivers the most consistent performance among the evaluated methods. Its learned graphs reflect task-specific sensor relationships, while graph sparsity varies with tuning and sample splits.

B. Real Data (O2)

  • The six real datasets cover brain-age regression from cortical thickness and wearable-sensor human-activity classification.The brain datasets are ADNI1, ADNI2, PPMI, and ABIDE; the motion datasets are MHEALTH and REALDISP.Source: B. Real Data (O2) · p. 8
  • PNN-Joint delivers the most consistent performance across real datasets, outperforms PNN-Naive on every dataset, and generally outperforms LDS.The table reports regression MAE, classification accuracy, and percentage of non-zero precision entries; NZ for PNN-Joint is computed on ˜Θ.Source: B. Real Data (O2) · p. 8, Table I · p. 9
  • PNN-Joint graphs are generally sparse but remain dense on some datasets because the main experiments tune hyperparameters for task performance rather than sparsity.On ADNI2, an alternative configuration yields 33% ± 37% non-zero entries with MAE of 5.36 ± 0.28.Source: B. Real Data (O2) · p. 8
  • On REALDISP, learned sensor edges vary with the task: leg-sensor connections appear for leg-relevant actions, and PNN-Joint adds more cross-body edges.For Task 1, PNN-Joint connects the chest to one upper-leg sensor; for Task 2, both methods estimate edges across leg sensors.Source: Figure 5 · p. 9
  • Learned graph sparsity varies across data splits, while task performance remains stable across runs.The authors suggest graphs with different sparsity levels can support the downstream task equally well.Source: B. Real Data (O2) · p. 9

C. Stability (O3)

PNN variants retain stability under finite-sample perturbations, while learned REALDISP graphs reflect task-specific sensor relationships. The section also positions PNN-Joint among graph-learning methods by its statistical grounding and trainable task model.

C. Stability (O3)

  • PNN-GL and PNN-Joint remain stable with fewer samples, and all PNNs are more stable than PCA under finite-sample errors.PNN-Sample suffers more in low-data regimes as inversion amplifies spurious noise; PNN-Joint achieves similar or better stability than PNN-GL despite worst-case bounds suggesting lower stability.Source: C. Stability (O3) · pp. 9–10

D. Interpretability (O4)

  • For REALDISP upper-body actions, PNN-GL estimates no leg-sensor edges while PNN-Joint links the chest to one upper-leg sensor; for lower-body actions, both connect leg sensors.PNN-Joint also estimates more edges, including leg-to-upper-body links; plotted body-location weights average corresponding sensor-node edges and use a 0.2 threshold.Source: D. Interpretability (O4) · p. 10

VII. RELATED WORK

  • Prior joint graph-learning methods use varied graph models and optimization schemes, but the cited approaches do not exploit graph structure or nonsmooth sparsity penalties and often lack statistical guarantees.The discussion also notes that these graph-estimation processes are not grounded in observed-data moments or covariance structures.Source: §VII · p. 10
  • PNN-Joint combines a trainable GNN task with a graph estimate anchored to the GMRF likelihood, unlike task-driven graph-recovery methods whose task is not a learned model.The section contrasts this with covariance-based VNN methods, whose regularized covariance estimates do not directly account for downstream labels.Source: §VII · p. 10

VIII. CONCLUSION

  • PNN-Joint showed superior predictive performance, low-data robustness, and interpretable graphs aligned with domain logic across synthetic, neuroimaging, and motion-sensor evaluations.Source: §VIII · p. 10

APPENDIX A AUXILIARY LEMMA

The appendix first gives a Lipschitz-filter perturbation lemma, then uses it in a finite-sample analysis of the sample precision estimator.

APPENDIX A AUXILIARY LEMMA

  • For symmetric matrices with spectra in an interval where the filter response is Lipschitz, the lemma bounds filtered-matrix differences by matrix perturbations and transfers the bound to filtered signals.Source: Appendix A · p. 12

APPENDIX B PROOF OF PROPOSITION 1

  • When T ≥ N, the sample covariance is invertible with probability one, so its inverse defines the sample precision estimate.Source: Appendix B · p. 12
  • The sample-precision proof bounds inverse-covariance error using an eigenvalue perturbation condition and a non-asymptotic covariance-error bound for sub-Gaussian data.The stated condition is that the covariance perturbation norm is at most half the minimum eigenvalue.Source: Appendix B · p. 12

APPENDIX C PROOF OF PROPOSITION 2

The proof of Proposition 2 applies a filter perturbation bound to graphical-lasso precision estimates and invokes a standard graphical-lasso error result.

APPENDIX C PROOF OF PROPOSITION 2

  • The Lipschitz-filter lemma transfers graphical-lasso precision-estimation error into a bound on the filtered signal.Source: Appendix C · p. 12
  • The cited graphical-lasso result gives an error rate proportional to ((N + S) log N)/T, with probability approaching one as N and T grow.Source: Appendix C · p. 12

APPENDIX D PROOF OF THEOREM 1

The proof analyzes the precision-matrix optimization for a fixed network estimate using convex optimality conditions. It combines bounds on the objective terms to obtain the theorem’s error bound.

APPENDIX D PROOF OF THEOREM 1

  • For fixed network estimate, the proof treats the precision loss as convex over a positive-definite feasible set with a spectral-norm bound and applies first-order optimality.Source: Appendix D · p. 12
  • The subgradient combines the likelihood, graphical-lasso sparsity penalty on off-diagonal entries, and a quadratic term measuring distance from the network estimate.Source: Appendix D · pp. 12–13
  • The proof combines likelihood, sparsity, and perturbation bounds, then uses the assumed distance between the precision and network estimates to obtain the final error bound.Source: Appendix D · p. 13

APPENDIX E PROOF OF THEOREM 2

The proof bounds the change in H applied to x using Lemma 1, then applies the triangle inequality and Theorem 1’s error estimate to establish the claim.

  • Lemma 1 bounds the difference between H(˜Θ(i))x and H(Θ0)x by a constant times ||˜Θ(i)−Θ0||F||x||2.Source: Appendix E · p. 13
  • The proof next applies the triangle inequality to relate the terms in the bound.Source: Appendix E · p. 13
  • Substituting the error bound from Theorem 1 yields an O_P expression for ||˜Θ(i)−Θ0||F, and the proof concludes that the claim follows.Source: Appendix E · p. 13

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