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Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrt{n}$ Dimension Threshold

Venkatesan Guruswami, Jun-Ting Hsieh, Prasad Raghavendra

TL;DR

This work addresses certifying that a random $d$-dimensional subspace of $\mathbb{R}^n$ is well-spread, i.e., has distortion $\Delta(X) \le O(1)$, in a regime where $d$ can exceed the classic $\sqrt{n}$ barrier. It introduces a subexponential-time certification framework that interpolates between the polynomial-time $d\le\tilde{O}(\sqrt{n})$ regime and the information-hard regime, via a novel combination of the $2\to 4$ norm proxy, trimming of large coordinates, and control of elementary symmetric polynomials $P_t$. The same machinery extends to the planted sparse-vector problem, offering a subexponential-time algorithm that recovers a sparse planted vector under a random subspace model by leveraging Sum-of-Squares and a careful rounding strategy. Collectively, the results establish a smooth runtime-dimension trade-off, provide subexponential algorithms in a hard regime, and connect certification techniques to planted recovery through SoS-based rounding.

Abstract

We consider the task of certifying that a random $d$-dimensional subspace $X$ in $\mathbb{R}^n$ is well-spread - every vector $x \in X$ satisfies $c\sqrt{n} \|x\|_2 \leq \|x\|_1 \leq \sqrt{n}\|x\|_2$. In a seminal work, Barak et. al. showed a polynomial-time certification algorithm when $d \leq O(\sqrt{n})$. On the other hand, when $d \gg \sqrt{n}$, the certification task is information-theoretically possible but there is evidence that it is computationally hard [MW21,Cd22], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the $d \gg \sqrt{n}$ regime. Our algorithm runs in time $\exp(\widetilde{O}(n^{\varepsilon}))$ when $d \leq \widetilde{O}(n^{(1+\varepsilon)/2})$, establishing a smooth trade-off between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension trade-off for this task.

Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrt{n}$ Dimension Threshold

TL;DR

This work addresses certifying that a random -dimensional subspace of is well-spread, i.e., has distortion , in a regime where can exceed the classic barrier. It introduces a subexponential-time certification framework that interpolates between the polynomial-time regime and the information-hard regime, via a novel combination of the norm proxy, trimming of large coordinates, and control of elementary symmetric polynomials . The same machinery extends to the planted sparse-vector problem, offering a subexponential-time algorithm that recovers a sparse planted vector under a random subspace model by leveraging Sum-of-Squares and a careful rounding strategy. Collectively, the results establish a smooth runtime-dimension trade-off, provide subexponential algorithms in a hard regime, and connect certification techniques to planted recovery through SoS-based rounding.

Abstract

We consider the task of certifying that a random -dimensional subspace in is well-spread - every vector satisfies . In a seminal work, Barak et. al. showed a polynomial-time certification algorithm when . On the other hand, when , the certification task is information-theoretically possible but there is evidence that it is computationally hard [MW21,Cd22], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the regime. Our algorithm runs in time when , establishing a smooth trade-off between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension trade-off for this task.
Paper Structure (25 sections, 23 theorems, 57 equations, 2 figures, 2 algorithms)

This paper contains 25 sections, 23 theorems, 57 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Finding planted sparse vector in a random subspace.
  3. Technical Overview
  4. Notation.
  5. Organization.
  6. 2-to-4 norm as a proxy for sparsity: certification algorithm of [BBH+12]
  7. Going beyond sqrt(n)
  8. Elementary symmetric polynomials
  9. Overview of the trace moment method
  10. Upper bound on Pt(x)
  11. Finding planted sparse vector
  12. Certifying Spread of a Random Subspace
  13. Removing the top entries of Ax
  14. Symmetrization lowers spectral norm
  15. Encoding of walks in the trace
  16. ...and 10 more sections

Key Result

Theorem 1

Fix $\varepsilon \in (0,1)$, and let $d, n \in \mathbb{N}$ such that $d = O(n^{\frac{1+\varepsilon}{2}} / \log n)$. Let $A \sim \mathcal{N}(0,1)^{n \times d}$. Then, there is a certification algorithm that runs in $2^{\widetilde{O}(n^{\varepsilon})}$ time and, with probability $1- o(1)$ over $A$, ce

Figures (2)

  • Figure 1: Examples of different structures of walks in the trace.
  • Figure 2: An example where we decode the same encoding with different structures.

Theorems & Definitions (51)

  • Theorem 1: Informal \ref{['thm:spread']}
  • Definition 1.1: Noisy Bernoulli-Rademacher distribution ChendO22
  • Theorem 2: Informal \ref{['thm:planted-sparse-vector']}
  • Remark 1.2
  • Lemma 2.1: Informal \ref{['lem:2-to-4-excluding-top']}
  • Lemma 2.2: Informal \ref{['lem:sum-of-distinct-products']}
  • Lemma 2.3: Informal \ref{['lem:trace-bound']}
  • Definition 3.1: Spreadness property of a subspace
  • Lemma 3.2: Lemma 2.11 of GuruswamiLR10
  • Theorem 3.3: Formal version of \ref{['thm:certification']}
  • ...and 41 more