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Detection Is Harder Than Estimation in Certain Regimes: Inference for Moment and Cumulant Tensors

Runshi Tang, Yuefeng Han, Anru R. Zhang

Abstract

We study estimation and detection of high-order moment and cumulant tensors from $n$ i.i.d. observations of a $p$-dimensional random vector, with performance measured in tensor spectral norm. On the statistical side, we prove that under sub-Gaussianity, the minimax rate for estimating the order-$d$ moment and cumulant tensors is $\sqrt{p/n}\wedge 1$. In contrast to covariance estimation, the sample moment tensor is generally no longer rate-optimal for higher-order moments. We therefore develop an estimator that attains the minimax rate up to logarithmic factors through a convex feasibility formulation over an $\varepsilon$-net of the unit sphere. On the computational side, we study the problem of testing whether the $d$-th order cumulant tensor vanishes after whitening. Using the low-degree polynomial framework, we provide evidence that detection is computationally hard when $n\ll p^{d/2}$. At the same time, we identify a regime in which an efficiently computable estimator attains error smaller than the separation at which low-degree tests can reliably distinguish the null from the alternative. This yields the striking conclusion that computationally efficient detection can be harder than computationally efficient estimation, revealing an unusual reverse detection-estimation gap: in a broad regime, computationally efficient estimation is possible at a smaller scale than computationally efficient detection. This phenomenon arises because the computational difficulty is driven not only by the statistical model, but also by the loss function itself: tensor spectral norm is NP-hard to compute. This feature makes the proposed open problems regarding computational lower bounds for estimation qualitatively different from the existing literature. Our results therefore uncover a new kind of computational--statistical gap.

Detection Is Harder Than Estimation in Certain Regimes: Inference for Moment and Cumulant Tensors

Abstract

We study estimation and detection of high-order moment and cumulant tensors from i.i.d. observations of a -dimensional random vector, with performance measured in tensor spectral norm. On the statistical side, we prove that under sub-Gaussianity, the minimax rate for estimating the order- moment and cumulant tensors is . In contrast to covariance estimation, the sample moment tensor is generally no longer rate-optimal for higher-order moments. We therefore develop an estimator that attains the minimax rate up to logarithmic factors through a convex feasibility formulation over an -net of the unit sphere. On the computational side, we study the problem of testing whether the -th order cumulant tensor vanishes after whitening. Using the low-degree polynomial framework, we provide evidence that detection is computationally hard when . At the same time, we identify a regime in which an efficiently computable estimator attains error smaller than the separation at which low-degree tests can reliably distinguish the null from the alternative. This yields the striking conclusion that computationally efficient detection can be harder than computationally efficient estimation, revealing an unusual reverse detection-estimation gap: in a broad regime, computationally efficient estimation is possible at a smaller scale than computationally efficient detection. This phenomenon arises because the computational difficulty is driven not only by the statistical model, but also by the loss function itself: tensor spectral norm is NP-hard to compute. This feature makes the proposed open problems regarding computational lower bounds for estimation qualitatively different from the existing literature. Our results therefore uncover a new kind of computational--statistical gap.

Paper Structure

This paper contains 48 sections, 20 theorems, 308 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of contributions.
  3. Related Work
  4. Independent component analysis (ICA)
  5. Tensor PCA
  6. Robust covariance estimation
  7. Polynomial chaos and concentration
  8. Moment methods
  9. Computational hardness
  10. Preliminaries and Notation
  11. Notation
  12. Moment and cumulant tensors
  13. Statistical Limit
  14. Lower Bounds for Estimation and Detection
  15. Sample Moment Tensors Can Be Suboptimal
  16. ...and 33 more sections

Key Result

Theorem 1

For fixed order $d\ge 3$, the problem of estimating and detecting high-order moment and cumulant tensors under tensor spectral norm exhibits the following behavior:

Figures (1)

  • Figure 1: Phase diagram of statistical and computational limits in moment and cumulant tensor estimation and detection

Theorems & Definitions (41)

  • Theorem 1: Informal
  • Theorem 2: Lower bound for estimating $d$-th order cumulant and moment tensors
  • Theorem 3: Statistical lower bound for detection
  • Remark 1: Detection is trivial for even-order moment tensors.
  • Theorem 4: Deviation bound for sample moment tensors
  • Theorem 5: Estimation of a multilinear form
  • Remark 2: Heuristic for the $\sqrt{p/n}$ rate
  • Theorem 6: Convex formulation for a statistically optimal estimator
  • Lemma 1: LDP norm upper bound
  • Theorem 7: Low-degree hardness for detecting a nonzero $d$-th order cumulant
  • ...and 31 more