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1. Lattice-Based Methods Surpass Sum-of-Squares in Clustering

   Ilias Zadik ; Min Jae Song ; Alexander S. Wein ; Joan Bruna ( July 2022 , conference on learning theory)

   Clustering is a fundamental primitive in unsupervised learning which gives rise to a rich class of computationally-challenging inference tasks. In this work, we focus on the canonical task of clustering d-dimensional Gaussian mixtures with unknown (and possibly degenerate) covariance. Recent works (Ghosh et al. ’20; Mao, Wein ’21; Davis, Diaz, Wang ’21) have established lower bounds against the class of low-degree polynomial methods and the sum-of-squares (SoS) hierarchy for recovering certain hidden structures planted in Gaussian clustering instances. Prior work on many similar inference tasks portends that such lower bounds strongly suggest the presence of an inherent statistical-to-computational gap formore »clustering, that is, a parameter regime where the clustering task is statistically possible but no polynomial-time algorithm succeeds. One special case of the clustering task we consider is equivalent to the problem of finding a planted hypercube vector in an otherwise random subspace. We show that, perhaps surprisingly, this particular clustering model does not exhibit a statistical-to-computational gap, despite the aforementioned low-degree and SoS lower bounds. To achieve this, we give an algorithm based on Lenstra–Lenstra–Lovász lattice basis reduction which achieves the statistically-optimal sample complexity of d + 1 samples. This result extends the class of problems whose conjectured statistical-to-computational gaps can be “closed” by “brittle” polynomial-time algorithms, highlighting the crucial but subtle role of noise in the onset of statistical-to-computational gaps.« less
2. Computational Hardness of Certifying Bounds on Constrained PCA Problems

   https://doi.org/10.4230/LIPIcs.ITCS.2020.78  

   Bandeira, Afonso S ; Kunisky, Dmitriy ; Wein, Alexander S ( June 2020 , 11th Innovations in Theoretical Computer Science Conference (ITCS 2020))

   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^⊤ 𝑾 𝒙 over all vectors 𝒙 in a constraint set 𝒮 ⊂ ℝⁿ. For a certain class of normalized constraint sets we show that, conditional on a certain complexity-theoretic conjecture, no polynomial-time algorithm can certify 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 certifyingmore »bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. Our results suggest a striking gap between optimization and certification for this problem. 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 predicting computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is conjectured 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.« less
3. Lower bounds for testing graphical models: colorings and antiferromagnetic Ising models

   Bezakova, Ivona ; Blanca, Antonio ; Chen, Zongchen ; Stefankovic, Daniel ; Vigoda, Eric ( July 2019 , Proceedings of Machine Learning Research)

   We study the identity testing problem in the context of spin systems or undirected graphical models, where it takes the following form: given the parameter specification of the model M and a sampling oracle for the distribution \mu_{M^*} of an unknown model M^*, can we efficiently determine if the two models M and M^* are the same? We consider identity testing for both soft-constraint and hard-constraint systems. In particular, we prove hardness results in two prototypical cases, the Ising model and proper colorings, and explore whether identity testing is any easier than structure learning. For the ferromagnetic (attractive) Ising model,more »Daskalasis et al. (2018) presented a polynomial time algorithm for identity testing. We prove hardness results in the antiferromagnetic (repulsive) setting in the same regime of parameters where structure learning is known to require a super-polynomial number of samples. In particular, for n-vertex graphs of maximum degree d, we prove that if |\beta| d = \omega(\log n) (where \beta is the inverse temperature parameter), then there is no identity testing algorithm for the antiferromagnetic Ising model that runs in polynomial time unless RP = NP. We also establish computational lower bounds for a broader set of parameters under the (randomized) exponential time hypothesis. In our proofs, we use random graphs as gadgets; this is inspired by similar constructions in seminal works on the hardness of approximate counting. In the hard-constraint setting, we present hardness results for identity testing for proper colorings. Our results are based on the presumed hardness of #BIS, the problem of (approximately) counting independent sets in bipartite graphs. In particular, we prove that identity testing for colorings is hard in the same range of parameters where structure learning is known to be hard, which in turn matches the parameter regime for NP-hardness of the corresponding decision problem.« less
4. Sublinear-Time Algorithms for Counting Star Subgraphs via Edge Sampling

   https://doi.org/10.1007/s00453-017-0287-3  

   Aliakbarpour, Maryam ; Biswas, Amartya Shankha ; Gouleakis, Themis ; Peebles, John ; Rubinfeld, Ronitt ; Yodpinyanee, Anak ( January 2018 , Algorithmica)

   We study the problem of estimating the value of sums of the form Sp≜∑(xip) when one has the ability to sample xi≥0 with probability proportional to its magnitude. When p=2, this problem is equivalent to estimating the selectivity of a self-join query in database systems when one can sample rows randomly. We also study the special case when {xi} is the degree sequence of a graph, which corresponds to counting the number of p-stars in a graph when one has the ability to sample edges randomly. Our algorithm for a (1±ε)-multiplicative approximation of Sp has query and time complexities O(mloglognϵ2S1/pp).more »Here, m=∑xi/2 is the number of edges in the graph, or equivalently, half the number of records in the database table. Similarly, n is the number of vertices in the graph and the number of unique values in the database table. We also provide tight lower bounds (up to polylogarithmic factors) in almost all cases, even when {xi} is a degree sequence and one is allowed to use the structure of the graph to try to get a better estimate. We are not aware of any prior lower bounds on the problem of join selectivity estimation. For the graph problem, prior work which assumed the ability to sample only vertices uniformly gave algorithms with matching lower bounds (Gonen et al. in SIAM J Comput 25:1365–1411, 2011). With the ability to sample edges randomly, we show that one can achieve faster algorithms for approximating the number of star subgraphs, bypassing the lower bounds in this prior work. For example, in the regime where Sp≤n, and p=2, our upper bound is O~(n/S1/2p), in contrast to their Ω(n/S1/3p) lower bound when no random edge queries are available. In addition, we consider the problem of counting the number of directed paths of length two when the graph is directed. This problem is equivalent to estimating the selectivity of a join query between two distinct tables. We prove that the general version of this problem cannot be solved in sublinear time. However, when the ratio between in-degree and out-degree is bounded—or equivalently, when the ratio between the number of occurrences of values in the two columns being joined is bounded—we give a sublinear time algorithm via a reduction to the undirected case.« less
5. Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs

   Afonso S Bandeira, Jess Banks ( October 2021 , Proceedings of Machine Learning Research)

   Abstract We study the problem of efficiently refuting the k-colorability of a graph, or equivalently, certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, suggesting that there is no polynomial-time algorithm that improves upon a classical spectral algorithm. Our evidence takes the form of a "computationally-quiet planting": we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. Wemore »generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model.« less