51 research outputs found
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. We prove that our algorithm achieves exact recovery of the
latent communities, up to the information-theoretic limits determined by Abbe
and Sandon (2015). Our result extends prior SDP approaches by allowing for many
communities of different sizes. By virtue of a semidefinite approach, our
algorithms succeed against a semirandom variant of the stochastic block model,
guaranteeing a form of robustness and generalization. We further explore how
semirandom models can lend insight into both the strengths and limitations of
SDPs in this setting.Comment: 29 page
Computational Hardness of Certifying Bounds on Constrained PCA Problems
Given a random n×n symmetric matrix W drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form x⊤Wx over all vectors x in a constraint set S⊂Rn. For a certain class of normalized constraint sets S we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of W. A notable special case included in our results is the hypercube S={±1/n−−√}n, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics.
Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over x∈S is much larger than that of a GOE matrix.ISSN:1868-896
Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials
Suppose we are given an -dimensional order-3 symmetric tensor that is the sum of random rank-1 terms. The
problem of recovering the rank-1 components is possible in principle when but polynomial-time algorithms are only known in the regime . Similar "statistical-computational gaps" occur in many
high-dimensional inference tasks, and in recent years there has been a flurry
of work on explaining the apparent computational hardness in these problems by
proving lower bounds against restricted (yet powerful) models of computation
such as statistical queries (SQ), sum-of-squares (SoS), and low-degree
polynomials (LDP). However, no such prior work exists for tensor decomposition,
largely because its hardness does not appear to be explained by a "planted
versus null" testing problem.
We consider a model for random order-3 tensor decomposition where one
component is slightly larger in norm than the rest (to break symmetry), and the
components are drawn uniformly from the hypercube. We resolve the computational
complexity in the LDP model: -degree polynomial functions of the
tensor entries can accurately estimate the largest component when but fail to do so when . This provides rigorous
evidence suggesting that the best known algorithms for tensor decomposition
cannot be improved, at least by known approaches. A natural extension of the
result holds for tensors of any fixed order , in which case the LDP
threshold is .Comment: 42 pages; STOC 202
Computational Barriers to Estimation from Low-Degree Polynomials
One fundamental goal of high-dimensional statistics is to detect or recover
structure from noisy data. In many cases, the data can be faithfully modeled by
a planted structure (such as a low-rank matrix) perturbed by random noise. But
even for these simple models, the computational complexity of estimation is
sometimes poorly understood. A growing body of work studies low-degree
polynomials as a proxy for computational complexity: it has been demonstrated
in various settings that low-degree polynomials of the data can match the
statistical performance of the best known polynomial-time algorithms for
detection. While prior work has studied the power of low-degree polynomials for
the task of detecting the presence of hidden structures, it has failed to
address the estimation problem in settings where detection is qualitatively
easier than estimation.
In this work, we extend the method of low-degree polynomials to address
problems of estimation and recovery. For a large class of "signal plus noise"
problems, we give a user-friendly lower bound for the best possible mean
squared error achievable by any degree-D polynomial. To our knowledge, this is
the first instance in which the low-degree polynomial method can establish
low-degree hardness of recovery problems where the associated detection problem
is easy. As applications, we give a tight characterization of the low-degree
minimum mean squared error for the planted submatrix and planted dense subgraph
problems, resolving (in the low-degree framework) open problems about the
computational complexity of recovery in both cases.Comment: 38 page
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. This conjecture formalizes the beliefs surrounding a line of recent work that seeks to understand statistical-versus-computational tradeoffs via the low-degree likelihood ratio. In this work, we refute the conjecture of Hopkins. However, our counterexample crucially exploits the specifics of the noise operator used in the conjecture, and we point out a simple way to modify the conjecture to rule out our counterexample. We also give an example illustrating that (even after the above modification), the symmetry assumption in the conjecture is necessary. These results do not undermine the low-degree framework for computational lower bounds, but rather aim to better understand what class of problems it is applicable to
Estimation under group actions: recovering orbits from invariants
Motivated by geometric problems in signal processing, computer vision, and
structural biology, we study a class of orbit recovery problems where we
observe very noisy copies of an unknown signal, each acted upon by a random
element of some group (such as Z/p or SO(3)). The goal is to recover the orbit
of the signal under the group action in the high-noise regime. This generalizes
problems of interest such as multi-reference alignment (MRA) and the
reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain
matching lower and upper bounds on the sample complexity of these problems in
high generality, showing that the statistical difficulty is intricately
determined by the invariant theory of the underlying symmetry group.
In particular, we determine that for cryo-EM with noise variance
and uniform viewing directions, the number of samples required scales as
. We match this bound with a novel algorithm for ab initio
reconstruction in cryo-EM, based on invariant features of degree at most 3. We
further discuss how to recover multiple molecular structures from heterogeneous
cryo-EM samples.Comment: 54 pages. This version contains a number of new result
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