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Sample Efficient Learning of Factored Embeddings of Tensor Fields

Taemin Heo, Chandrajit Bajaj

TL;DR

The paper tackles the challenge of processing enormous tensor fields by introducing Progressive Sketching (P‑SCT), a sample‑efficient framework that learns a sub‑sampling policy to build compact, latent factored embeddings of tensors. It combines Dirichlet‑based Thompson sampling with a SAD‑driven information measure to selectively sample tensor slices, achieving a rank‑$r$ Tucker approximation with reduced data usage. Theoretical guarantees extend SketchyCoreSVD to multi‑dimensional unfoldings, delivering bounded errors with high probability, and empirical results on Cardiac MRI, NARR air temperature, and hyperspectral imagery show faster convergence and competitive accuracy using roughly half the data compared with full‑scan baselines. These contributions offer scalable, accurate tensor embeddings for large scientific datasets, enabling efficient querying and post‑processing in resource‑constrained settings.

Abstract

Data tensors of orders 2 and greater are now routinely being generated. These data collections are increasingly huge and growing. Many scientific and medical data tensors are tensor fields (e.g., images, videos, geographic data) in which the spatial neighborhood contains important information. Directly accessing such large data tensor collections for information has become increasingly prohibitive. We learn approximate full-rank and compact tensor sketches with decompositive representations providing compact space, time and spectral embeddings of tensor fields. All information querying and post-processing on the original tensor field can now be achieved more efficiently and with customizable accuracy as they are performed on these compact factored sketches in latent generative space. We produce optimal rank-r sketchy Tucker decomposition of arbitrary order data tensors by building compact factor matrices from a sample-efficient sub-sampling of tensor slices. Our sample efficient policy is learned via an adaptable stochastic Thompson sampling using Dirichlet distributions with conjugate priors.

Sample Efficient Learning of Factored Embeddings of Tensor Fields

TL;DR

The paper tackles the challenge of processing enormous tensor fields by introducing Progressive Sketching (P‑SCT), a sample‑efficient framework that learns a sub‑sampling policy to build compact, latent factored embeddings of tensors. It combines Dirichlet‑based Thompson sampling with a SAD‑driven information measure to selectively sample tensor slices, achieving a rank‑ Tucker approximation with reduced data usage. Theoretical guarantees extend SketchyCoreSVD to multi‑dimensional unfoldings, delivering bounded errors with high probability, and empirical results on Cardiac MRI, NARR air temperature, and hyperspectral imagery show faster convergence and competitive accuracy using roughly half the data compared with full‑scan baselines. These contributions offer scalable, accurate tensor embeddings for large scientific datasets, enabling efficient querying and post‑processing in resource‑constrained settings.

Abstract

Data tensors of orders 2 and greater are now routinely being generated. These data collections are increasingly huge and growing. Many scientific and medical data tensors are tensor fields (e.g., images, videos, geographic data) in which the spatial neighborhood contains important information. Directly accessing such large data tensor collections for information has become increasingly prohibitive. We learn approximate full-rank and compact tensor sketches with decompositive representations providing compact space, time and spectral embeddings of tensor fields. All information querying and post-processing on the original tensor field can now be achieved more efficiently and with customizable accuracy as they are performed on these compact factored sketches in latent generative space. We produce optimal rank-r sketchy Tucker decomposition of arbitrary order data tensors by building compact factor matrices from a sample-efficient sub-sampling of tensor slices. Our sample efficient policy is learned via an adaptable stochastic Thompson sampling using Dirichlet distributions with conjugate priors.
Paper Structure (22 sections, 1 theorem, 9 equations, 13 figures, 2 tables, 3 algorithms)

This paper contains 22 sections, 1 theorem, 9 equations, 13 figures, 2 tables, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND
  3. Tucker Decomposition and Higher Order SVD (HOSVD)
  4. Thompson Sampling (TS)
  5. Uniqueness of R-SCT and P-SCT
  6. PROGRESSIVE SKETCHING
  7. Progressive SketchyCoreTucker (P-SCT)
  8. Complexity Analysis
  9. Theoretical Guarantees
  10. EXPERIMENTS
  11. Cardiac Magnetic Resonance Imaging
  12. NARR Air Temperature
  13. Hyperspectral Image
  14. Discussion
  15. CONCLUSION
  16. ...and 7 more sections

Key Result

Theorem 1

Every order 3 tensor $\mathcal{A}$ can be written as the product in which $\bm{U}^{(k)}$ is a unitary $(n_k\times n_k)$-matrix; $\mathcal{S}$ is a $(n_1\times n_2\times n_3)$-tensor of which subtensors $\mathcal{S}_{i_k=\alpha}$, obtained by fixing the kth index to $\alpha$, have the property of all-orthogonality and ordering for all possible k.

Figures (13)

  • Figure 1: Performance comparison for Cardiac MRI dataset. Violin plots of error and computation time (sec) from 100 trials. (see Table \ref{['table:result']} for more information).
  • Figure 2: Learning curve comparison. P-SCT (orange curves) converges faster than R-SCT (blue curves), showing sample-efficient behavior. P-SCT's error is smaller than R-SCT with smaller variances in almost every trial, empirically showing P-SCT's tighter error bounds. P-SCT produces a more accurate tensor decomposition than most full-scan algorithms using less than half of the input tensor.
  • Figure 3: Performance comparison for NARR air temperature dataset. Violin plots of error and computation time (sec) from 100 trials. (see Table \ref{['table:result']} for more information).
  • Figure 4: Learning curve comparison. P-SCT (orange curves) converges faster than R-SCT (blue curves), showing sample-efficient behavior. P-SCT's error is smaller than R-SCT with smaller variances in almost every trial, empirically showing P-SCT's tighter error bounds. P-SCT produces a more accurate tensor decomposition than most full-scan algorithms using less than half of the input tensor.
  • Figure 5: Performance comparison for hyperspectral image. Violin plots of error and computation time (sec) from 100 trials. (see Table \ref{['table:result']} for more information).
  • ...and 8 more figures

Theorems & Definitions (1)

  • Theorem 1: see de2000multilinear