Search Results for author:
Found 30 papers, 0 papers with code
Low-degree phase transitions for detecting a planted clique in sublinear time
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.
Information-Theoretic Thresholds for Planted Dense Cycles
We study a random graph model for small-world networks which are ubiquitous in social and biological sciences.
Precise Error Rates for Computationally Efficient Testing
Our result shows that the spectrum is a sufficient statistic for computationally bounded tests (but not for all tests).
Detection of Dense Subhypergraphs by Low-Degree Polynomials
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 $0 < \gamma < 1$.
Detection-Recovery Gap for Planted Dense Cycles
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.
Is it easier to count communities than find them?
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.
Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials
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 $r \ll n^{3/2}$.
Near-optimal fitting of ellipsoids to random points
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 $n \sim d^2/4$.
Statistical and Computational Phase Transitions in Group Testing
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).
The Franz-Parisi Criterion and Computational Trade-offs in High Dimensional Statistics
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.
Lattice-Based Methods Surpass Sum-of-Squares in Clustering
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.
Optimal Spectral Recovery of a Planted Vector in a Subspace
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.
Optimal Low-Degree Hardness of Maximum Independent Set
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 $(\log d / d)n$.
Computational Barriers to Estimation from Low-Degree Polynomials
One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data.
Free Energy Wells and Overlap Gap Property in Sparse PCA
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.
The Average-Case Time Complexity of Certifying the Restricted Isometry Property
A matrix has the $(s,\delta)$-$\mathsf{RIP}$ property if behaves as a $\delta$-approximate isometry on $s$-sparse vectors.
Computationally efficient sparse clustering
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.
Hardness of Random Optimization Problems for Boolean Circuits, Low-Degree Polynomials, and Langevin Dynamics
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).
Counterexamples to the Low-Degree Conjecture
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.
Notes on Computational Hardness of Hypothesis Testing: Predictions using the Low-Degree Likelihood Ratio
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.
Subexponential-Time Algorithms for Sparse PCA
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 $\rho \lesssim 1/\sqrt{n}$.
The Kikuchi Hierarchy and Tensor PCA
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).
Spectral Methods from Tensor Networks
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.
Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models
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.
Notes on computational-to-statistical gaps: predictions using statistical physics
In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems.
Statistical limits of spiked tensor models
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 $d$.
Message-passing algorithms for synchronization problems over compact groups
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).
Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization
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.
How Robust are Reconstruction Thresholds for Community Detection?
The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection.
A semidefinite program for unbalanced multisection in the stochastic block model
We propose a semidefinite programming (SDP) algorithm for community detection in the stochastic block model, a popular model for networks with latent community structure.
