no code implementations • 29 Apr 2024 • Dmitriy Kunisky, Cristopher Moore, Alexander S. Wein

This basis lets us unify and strengthen previous results on low-degree hardness, giving a combinatorial explanation of the hardness transition and of a continuum of subexponential-time algorithms that work below it, and proving tight lower bounds against low-degree polynomials for recovering rather than just detecting the signal.

no code implementations • 8 Feb 2024 • Jay Mardia, Kabir Aladin Verchand, Alexander S. Wein

Using (a bound on) the conditional low-degree likelihood ratio as a potential function, we show that for every non-adaptive query pattern, there is a highly structured query pattern of the same size that is at least as effective.

no code implementations • 1 Feb 2024 • Cheng Mao, Alexander S. Wein, Shenduo Zhang

We study a random graph model for small-world networks which are ubiquitous in social and biological sciences.

no code implementations • 1 Nov 2023 • Ankur Moitra, Alexander S. Wein

Our result shows that the spectrum is a sufficient statistic for computationally bounded tests (but not for all tests).

no code implementations • 17 Apr 2023 • Abhishek Dhawan, Cheng Mao, Alexander S. Wein

We consider detecting the presence of a planted $G^r(n^\gamma, n^{-\alpha})$ subhypergraph in a $G^r(n, n^{-\beta})$ hypergraph, where $0< \alpha < \beta < r-1$ and $0 < \gamma < 1$.

no code implementations • 13 Feb 2023 • Cheng Mao, Alexander S. Wein, Shenduo Zhang

Planted dense cycles are a type of latent structure that appears in many applications, such as small-world networks in social sciences and sequence assembly in computational biology.

no code implementations • 21 Dec 2022 • Cynthia Rush, Fiona Skerman, Alexander S. Wein, Dana Yang

In particular, we consider certain hypothesis testing problems between models with different community structures, and we show (in the low-degree polynomial framework) that testing between two options is as hard as finding the communities.

no code implementations • 10 Nov 2022 • Alexander S. Wein

The problem of recovering the rank-1 components is possible in principle when $r \lesssim n^2$ but polynomial-time algorithms are only known in the regime $r \ll n^{3/2}$.

no code implementations • 19 Aug 2022 • Aaron Potechin, Paxton Turner, Prayaag Venkat, Alexander S. Wein

6031-6036, 2013] conjecture that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points $n$ increases, with a sharp threshold at $n \sim d^2/4$.

no code implementations • 15 Jun 2022 • Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Alexander S. Wein, Ilias Zadik

For the Bernoulli design, we determine the precise number of tests required to solve the associated detection problem (where the goal is to distinguish between a group testing instance and pure noise), improving both the upper and lower bounds of Truong, Aldridge, and Scarlett (2020).

no code implementations • 19 May 2022 • Afonso S. Bandeira, Ahmed El Alaoui, Samuel B. Hopkins, Tselil Schramm, Alexander S. Wein, Ilias Zadik

We define a free-energy based criterion for hardness and formally connect it to the well-established notion of low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal.

no code implementations • 7 Dec 2021 • Ilias Zadik, Min Jae Song, Alexander S. Wein, Joan Bruna

Prior work on many similar inference tasks portends that such lower bounds strongly suggest the presence of an inherent statistical-to-computational gap for clustering, that is, a parameter regime where the clustering task is statistically possible but no polynomial-time algorithm succeeds.

no code implementations • 31 May 2021 • Cheng Mao, Alexander S. Wein

Recovering a planted vector $v$ in an $n$-dimensional random subspace of $\mathbb{R}^N$ is a generic task related to many problems in machine learning and statistics, such as dictionary learning, subspace recovery, principal component analysis, and non-Gaussian component analysis.

no code implementations • 13 Oct 2020 • Alexander S. Wein

The maximum independent set is known to have size $(2 \log d / d)n$ in the double limit $n \to \infty$ followed by $d \to \infty$, but the best known polynomial-time algorithms can only find an independent set of half-optimal size $(\log d / d)n$.

no code implementations • 5 Aug 2020 • Tselil Schramm, Alexander S. Wein

One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data.

no code implementations • 18 Jun 2020 • Gérard Ben Arous, Alexander S. Wein, Ilias Zadik

We study a variant of the sparse PCA (principal component analysis) problem in the "hard" regime, where the inference task is possible yet no polynomial-time algorithm is known to exist.

no code implementations • 22 May 2020 • Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira

A matrix has the $(s,\delta)$-$\mathsf{RIP}$ property if behaves as a $\delta$-approximate isometry on $s$-sparse vectors.

no code implementations • 21 May 2020 • Matthias Löffler, Alexander S. Wein, Afonso S. Bandeira

We study statistical and computational limits of clustering when the means of the centres are sparse and their dimension is possibly much larger than the sample size.

no code implementations • 25 Apr 2020 • David Gamarnik, Aukosh Jagannath, Alexander S. Wein

For the case of Boolean circuits, our results improve the state-of-the-art bounds known in circuit complexity theory (although we consider the search problem as opposed to the decision problem).

no code implementations • 17 Apr 2020 • Justin Holmgren, Alexander S. Wein

A conjecture of Hopkins (2018) posits that for certain high-dimensional hypothesis testing problems, no polynomial-time algorithm can outperform so-called "simple statistics", which are low-degree polynomials in the data.

no code implementations • 26 Jul 2019 • Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira

Prior work has shown that when the signal-to-noise ratio ($\lambda$ or $\beta\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = \rho$, it is possible to recover $x$ in polynomial time if $\rho \lesssim 1/\sqrt{n}$.

no code implementations • 26 Jul 2019 • Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira

These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems.

no code implementations • 8 Apr 2019 • Alexander S. Wein, Ahmed El Alaoui, Cristopher Moore

Our hierarchy is analogous to the sum-of-squares (SOS) hierarchy but is instead inspired by statistical physics and related algorithms such as belief propagation and AMP (approximate message passing).

no code implementations • 2 Nov 2018 • Ankur Moitra, Alexander S. Wein

Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks.

no code implementations • 2 Jul 2018 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra

Our results leverage Le Cam's notion of contiguity, and include: i) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for certain natural priors for the spike.

no code implementations • 29 Mar 2018 • Afonso S. Bandeira, Amelia Perry, Alexander S. Wein

In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems.

no code implementations • 22 Dec 2016 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira

Finally, for priors (i) and (ii) we carry out the replica prediction from statistical physics, which is conjectured to give the exact information-theoretic threshold for any fixed $d$.

no code implementations • 14 Oct 2016 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra

Various alignment problems arising in cryo-electron microscopy, community detection, time synchronization, computer vision, and other fields fall into a common framework of synchronization problems over compact groups such as Z/L, U(1), or SO(3).

no code implementations • 19 Sep 2016 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra

Our results include: I) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for a variety of benign priors for the spike.

no code implementations • 4 Nov 2015 • Ankur Moitra, William Perry, Alexander S. Wein

The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection.

no code implementations • 20 Jul 2015 • Amelia Perry, Alexander S. Wein

We propose a semidefinite programming (SDP) algorithm for community detection in the stochastic block model, a popular model for networks with latent community structure.

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