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. We 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.

We propose a new, efficiently solvable hierarchy of semidefinite programming relaxations for inference problems. As test cases, we consider the problem of community detection in block models. The vertices are partitioned into k communities, and a graph is sampled conditional on a prescribed number of inter- and intra-community edges. The problem of detection, where we are to decide with high probability whether a graph was drawn from this model or the uniform distribution on regular graphs, is conjectured to undergo a computational phase transition at a point called the Kesten-Stigum (KS) threshold. In this work, we consider two models of random graphs namely the well-studied (irregular) Stochastic Block Model and a distribution over random regular graphs we'll call the Degree Regular Block Model. For both these models, we show that sufficiently high constant levels of our hierarchy can perform detection arbitrarily close to the KS threshold and that our algorithm is robust to up to a linear number of adversarial edge perturbations. Furthermore, in the case of Degree Regular Block Model, we show that below the Kesten-Stigum threshold no constant level can do so. In the case of the (irregular) Stochastic Block Model, it is known that efficient algorithms exist all the way down to this threshold, although none are robust to adversarial perturbation of a linear number of edges. More importantly, there is little complexity-theoretic evidence that detection is hard below the threshold. In the DRBM with more than two groups, it has not to our knowledge been proven that any algorithm succeeds down to the KS threshold, let alone that one can do so robustly, and there is a similar dearth of evidence for hardness below this point. Our SDP hierarchy is highly general and applicable to a wide range of hypothesis testing problems.

We exhibit a randomized algorithm which given a square matrix A ∈ \mathbbCn×nwith ||A|| ≤ 1 and , computes with high probability an invertible V and diagonal D such that ||A-VDV-1|| ≤ δ in O(TMM(n)log2(n/δ)) arithmetic operations on a floating point machine with O(log4(n/δ)logn) bits of precision. The computed similarity V additionally satisfies ||V||||V-1|| ≤ O(n2.5/δ). Here TMM(n) is the number of arithmetic operations required to multiply two n×n complex matrices numerically stably, known to satisfy TMM(n)=O(nω+η) for every where ω is the exponent of matrix multiplication [1]. The algorithm is a variant of the spectral bisection algorithm in numerical linear algebra [2] with a crucial Gaussian perturbation preprocessing step. Our running time is optimal up to polylogarithmic factors, in the sense that verifying that a given similarity diagonalizes a matrix requires at least matrix multiplication time. It significantly improves the previously best known provable running times of O(n10/δ2) arithmetic operations for diagonalization of general matrices [3], and (with regards to the dependence on n) O(n3) arithmetic operations for Hermitian matrices [4], and is the first algorithm to achieve nearly matrix multiplication time for diagonalization in any model of computation (real arithmetic, rational arithmetic, or finite arithmetic). The proof rests on two new ingredients. (1) We show that adding a small complex Gaussian perturbation to any matrix splits its pseudospectrum into n small well-separated components. In particular, this implies that the eigenvalues of the perturbed matrix have a large minimum gap, a property of independent interest in random matrix theory. (2) We give a rigorous analysis of Roberts' [5] Newton iteration method for computing the sign function of a matrix in finite arithmetic, itself an open problem in numerical analysis since at least 1986 [6]. This is achieved by controlling the evolution of the pseudospectra of the iterates using a carefully chosen sequence of shrinking contour integrals in the complex plane.

We derive upper and lower bounds on the degree $d$ for which the Lovász $\vartheta$ function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a $k$-coloring in random regular graphs $G_{n,d}$. We show that this type of refutation fails well above the $k$-colorability transition, and in particular everywhere below the Kesten--Stigum threshold. This is consistent with the conjecture that refuting $k$-colorability, or distinguishing $G_{n,d}$ from the planted coloring model, is hard in this region. Our results also apply to the disassortative case of the stochastic block model, adding evidence to the conjecture that there is a regime where community detection is computationally hard even though it is information-theoretically possible. Using orthogonal polynomials, we also provide explicit upper bounds on $\vartheta(\overline{G})$ for regular graphs of a given girth, which may be of independent interest.

We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these “planted models” and null models where the signal matrix is zero. For sparse PCA and submatrix localization, we determine this threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse; for the clustering problem, our bounds differ by a factor of $\sqrt {2}$ when the number of clusters is large. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information-theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured “hard but detectable” regime for community detection in sparse graphs.

In a paper that initiated the modern study of the stochastic block model, Decelle et al., backed by Mossel et al., made the following conjecture: Denote by $k$ the number of balanced communities, $a/n$ the probability of connecting inside communities and $b/n$ across, and set $\mathrm{SNR}=(a-b)^2/(k(a+(k-1)b)$; for any $k \geq 2$, it is possible to detect communities efficiently whenever $\mathrm{SNR}>1$ (the KS threshold), whereas for $k\geq 4$, it is possible to detect communities information-theoretically for some $\mathrm{SNR}<1$. Massoulié, Mossel et al.\ and Bordenave et al.\ succeeded in proving that the KS threshold is efficiently achievable for $k=2$, while Mossel et al.\ proved that it cannot be crossed information-theoretically for $k=2$. The above conjecture remained open for $k \geq 3$. This paper proves this conjecture, further extending the efficient detection to non-symmetrical SBMs with a generalized notion of detection and KS threshold. For the efficient part, a linearized acyclic belief propagation (ABP) algorithm is developed and proved to detect communities for any $k$ down to the KS threshold in time $O(n \log n)$. Achieving this requires showing optimality of ABP in the presence of cycles, a challenge for message passing algorithms. The paper further connects ABP to a power iteration method with a nonbacktracking operator of generalized order, formalizing the interplay between message passing and spectral methods. For the information-theoretic (IT) part, a non-efficient algorithm sampling a typical clustering is shown to break down the KS threshold at $k=4$. The emerging gap is shown to be large in some cases; if $a=0$, the KS threshold reads $b \gtrsim k^2$ whereas the IT bound reads $b \gtrsim k \ln(k)$, making the SBM a good study-case for information-computation gaps.

We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these “planted models” and null models where the signal matrix is zero. For sparse PCA and submatrix localization, we determine this threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse; for the clustering problem, our bounds differ by a factor √2 when the number of clusters is large. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information-theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured `hard but detectable' regime for community detection in sparse graphs.

We give upper and lower bounds on the information-theoretic threshold for community detection in the stochastic block model. Specifically, consider a symmetric stochastic block model with q groups, average degree d, and connection probabilities c_\mathrmin/n and c_\mathrmout/n for within-group and between-group edges respectively; let λ= (c_\mathrmin-c_\mathrmout)/(qd). We show that, when q is large, and λ= O(1/q), the critical value of d at which community detection becomes possible—in physical terms, the condensation threshold—is $ d_\mathrmc = Θ\left( \frac\log qq λ^2 \right) , with tighter results in certain regimes. Above this threshold, we show that any partition of the nodes into q groups which is as ‘good’ as the planted one, in terms of the number of within- and between-group edges, is correlated with it. This gives an exponential-time algorithm that performs better than chance; specifically, community detection becomes possible below the Kesten-Stigum bound for q \ge 5 in the disassortative case λ< 0, and for q \ge 11 in the assortative case λ> 0 (similar upper bounds were obtained independently by Abbe and Sandon). Conversely, below this threshold, we show that no algorithm can label the vertices better than chance, or even distinguish the block model from an Erdős-Rényi random graph with high probability. Our lower bound on d_\mathrmc uses Robinson and Wormald’s small subgraph conditioning method, and we also give (less explicit) results for non-symmetric stochastic block models. In the symmetric case, we obtain explicit results by using bounds on certain functions of doubly stochastic matrices due to Achlioptas and Naor; indeed, our lower bound on d_\mathrmc is their second moment lower bound on the q$-colorability threshold for random graphs with a certain effective degree.