In this work, we revisit algorithms for Tensor PCA: given an order-\(r\) tensor of the form \(T = G + \lambda \cdot v^{\otimes r}\) where \(G\) is a random symmetric Gaussian tensor with unit variance entries and \(v\) is an unknown boolean vector in \(\{\pm 1\}^n\), what's the minimum \(\lambda\) at which one can distinguish \(T\) from a random Gaussian tensor and more generally, recover \(v\)? As a result of a long line of work, we know that for any \(\ell \in \mathbb{N}\), there is a \(n^{O(\ell)}\) time algorithm that succeeds when the signal strength \(\lambda \gtrsim \sqrt{\log n} \cdot n^{-r/4} \cdot \ell^{1/2 - r/4}\). The question of whether the logarithmic factor is necessary turns out to be crucial to understanding whether larger polynomial time allows recovering the signal at a lower signal strength. Such a smooth trade-off is necessary for tensor PCA being a candidate problem for quantum speedups. It was first conjectured and then, more recently, with an eye on smooth trade-offs, reiterated in a blogpost of Bandeira.

In this work, we give novel spectral norm bounds for graph matrix on inputs being random regular graphs. Graph matrix is a family of random matrices with entries given by polynomial functions of the underlying input. These matrices have been known to be the backbone for the analysis of various average-case algorithms and hardness. Previous investigations of such matrices are largely restricted to the Erdős-Rényi model, and tight matrix norm bounds on regular graphs are only known for specific examples. We unite these two lines of investigations, and give the first result departing from the Erdős-Rényi setting in the full generality of graph matrices. We believe our norm bound result would enable a simple transfer of spectral analysis for average-case algorithms and hardness between these two distributions of random graphs. As an application of our spectral norm bounds, we show that higher-degree Sum-of-Squares lower bounds for the independent set problem on Erdős-Rényi random graphs can be switched into lower bounds on random d-regular graphs. Our main conceptual insight is that existing Sum-of-Squares lower bounds analysis based on moment methods are surprisingly robust, and amenable for a light-weight translation. Our result is the first to address the general open question of analyzing higher-degree Sum-of-Squares on random regular graphs.

We prove that for every D ∈ N, and large enough constant d ∈ N, with high probability over the choice of G ∼ G(n,d/n), the Erdos-Renyi random graph distribution, the canonical degree 2D Sum-of-Squares relaxation fails to certify that the largest independent set in G is of size o(n/√d D4). In particular, degree D sum-of-squares strengthening can reduce the integrality gap of the classical theta SDP relaxation by at most a O(D4) factor. This is the first lower bound for >4-degree Sum-of-Squares (SoS) relaxation for any problems on ultra sparse random graphs (i.e. average degree of an absolute constant). Such ultra-sparse graphs were a known barrier for previous methods and explicitly identified as a major open direction. Indeed, the only other example of an SoS lower bound on ultra-sparse random graphs was a degree-4 lower bound for Max-Cut. Our main technical result is a new method to obtain spectral norm estimates on graph matrices (a class of low-degree matrix-valued polynomials in G(n,d/n)) that are accurate to within an absolute constant factor. All prior works lose log n factors that trivialize any lower bound on o(logn)-degree random graphs. We combine these new bounds with several upgrades on the machinery for analyzing lower-bound witnesses constructed by pseudo-calibration so that our analysis does not lose any ω(1)-factors that would trivialize our results. In addition to other SoS lower bounds, we believe that our methods for establishing spectral norm estimates on graph matrices will be useful in the analyses of numerical algorithms on average-case inputs.

We study the computational complexity of two related problems: recovering a planted q-coloring in G(n,1/2), and finding efficiently verifiable witnesses of non-q-colorability (a.k.a. refutations) in G(n,1/2). Our main results show hardness for both these problems in a restricted-but-powerful class of algorithms based on computing low-degree polynomials in the inputs.The problem of recovering a planted q-coloring is equivalent to recovering q disjoint planted cliques that cover all the vertices — a potentially easier variant of the well-studied planted clique problem. Our first result shows that this variant is as hard as the original planted clique problem in the low-degree polynomial model of computation: each clique needs to have size k >> sqrt(n) for efficient recovery to be possible. For the related variant where the cliques cover a (1-epsilon)-fraction of the vertices, we also show hardness by reduction from planted clique.Our second result shows that refuting q-colorability of G(n,1/2) is hard in the low-degree polynomial model when q >> n^{2/3} but easy when q << n^{1/2}, and we leave closing this gap for future work. Our proof is more subtle than similar results for planted clique and involves constructing a non-standard distribution over q-colorable graphs. We note that while related to several prior works, this is the first work that explicitly formulates refutation problems in the low-degree polynomial model.The proofs of our main results involve showing low-degree hardness of hypothesis testing between an appropriately constructed pair of distributions. For refutation, we show completeness of this approach: in the low-degree model, the refutation task is precisely as hard as the hardest associated testing problem, i.e., proving hardness of refutation amounts to finding a "hard" distribution.

Given a graph and an integer k, Densest k-Subgraph is the algorithmic task of finding the subgraph on k vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an O(n1/4 + )-factor approximation (for any > 0) due to Bhaskara et al. [STOC ’10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an O(n1/4 − )-factor approximation. In the average case, Densest k-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap.

An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following question: What does the solution space for a random CSP look like to an efficient algorithm? In pursuit of this inquiry, we focus on the following problems about random Boolean CSPs at the densities where they are unsatisfiable but no refutation algorithm is known. 1) Counts. For every Boolean CSP we give algorithms that with high probability certify a subexponential upper bound on the number of solutions. We also give algorithms to certify a bound on the number of large cuts in a Gaussian-weighted graph, and the number of large independent sets in a random d-regular graph. 2) Clusters. For Boolean 3CSPs we give algorithms that with high probability certify an upper bound on the number of clusters of solutions. 3) Balance. We also give algorithms that with high probability certify that there are no "unbalanced" solutions, i.e., solutions where the fraction of +1s deviates significantly from 50%. Finally, we also provide hardness evidence suggesting that our algorithms for counting are optimal.

We give efficient algorithms for finding power-sum decomposition of an input polynomial $P(x)=\displaystyle \sum_{i\leq m}p_{i}(x)^{d}$ with component $p_{i}s$. The case of linear $p_{i}s$ is equivalent to the well-studied tensor decomposition problem while the quadratic case occurs naturally in studying identifiability of non-spherical Gaussian mixtures from low-order moments. Unlike tensor decomposition, both the unique identifiability and algorithms for this problem are not well-understood. For the simplest setting of quadratic $p_{i}s$ and $d=3$, prior work of [11] yields an algorithm only when $m\leq\overline{O}(\sqrt{n})$. On the other hand, the more general recent result of [13] builds an algebraic approach to handle any $m=n^{O(1)}$ components but only when d is large enough (while yielding no bounds for d=3 or even d=100) and only handles an inverse exponential noise. Our results obtain a substantial quantitative improvement on both the prior works above even in the base case of d=3 and quadratic $p_{i}s$. Specifically, our algorithm succeeds in decomposing a sum of $m\sim\overline{O}(n)$ generic quadratic $p_{i}s$ for $d=3$ and more generally the dth power-sum of $m\sim n^{2d/15}$ generic degree-K polynomials for any K$\geq$2. Our algorithm relies only on basic numerical linear algebraic primitives, is exact (i.e., obtain arbitrarily tiny error up to numerical precision), and handles an inverse polynomial noise when the $p_{i}s$ have random Gaussian coefficients. Our main tool is a new method for extracting the linear span of $p_{i}s$ by studying the linear subspace of low-order partial derivatives of the input P. For establishing polynomial stability of our algorithm in average-case, we prove inverse polynomial bounds on the smallest singular value of certain correlated random matrices with low-degree polynomial entries that arise in our analyses. Since previous techniques only yield significantly weaker bounds, we analyze the smallest singular value of matrices by studying the largest singular value of certain deviation matrices via graph matrix decomposition and the trace moment method.

The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong evidence of algorithmic hardness or information-computation gaps. Prior to this work, SoS lower bounds have been obtained for problems in the “dense” input regime, where the input is a collection of independent Rademacher or Gaussian random variables, while the sparse regime has remained out of reach. We make the first progress in this direction by obtaining strong SoS lower bounds for the problem of Independent Set on sparse random graphs. We prove that with high probability over an Erdós-Rénvi random graph$G\sim G_{n_{J}\frac{d}{u}}$with average degree$d > \log^{2}n$, degree-Dsos SoS fails to refute the existence of an independent set of size$k=\displaystyle \Omega(\frac{n}{\sqrt{d}(\log n)(\mathrm{D}_{\mathrm{S}\mathrm{o}\mathrm{S}})^{c_{0}}})$in$G$(where$c_{0}$is an absolute constant), whereas the true size of the largest independent set in$G$is$O(\displaystyle \frac{n\log d}{d})$. Our proof involves several significant extensions of the techniques used for proving SoS lower bounds in the dense setting. Previous lower bounds are based on the pseudo-calibration heuristic of Barak et al. [FOCS 2016] which produces a candidate SoS solution using a planted distribution indistinguishable from the input distribution via low-degree tests. In the sparse case the natural planted distribution does admit low-degree distinguishers, and we show how to adapt the pseudo-calibration heuristic to overcome this. Another notorious technical challenge for the sparse regime is the quest for matrix norm bounds. In this paper, we obtain new norm bounds for graph matrices in the sparse setting. While in the dense setting the norms of graph matrices are characterized by the size of the minimum vertex separator of the corresponding graph, this turns not to be the case for sparse graph matrices. Another contribution of our work is developing a new combinatorial understanding of structures needed to understand the norms of sparse graph matrices.

The degree-4 Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit algorithm with relatively low computational complexity, the limits of degree-4 SoS SDP are not well understood. For example, existing integrality gaps do not rule out a (2−)-algorithm for Vertex Cover or a (0.878+)-algorithm for MaxCut via degree-4 SoS SDPs, each of which would refute the notorious Unique Games Conjecture.

CMOS scaling so far enabled simultaneous system throughput scaling by concurrent improvements in delay, power, and area with thanks to Moore's law. CMOS scaling becomes more difficult with the limits of interconnect and increasing wafer cost. It is empirical to consider the system-on-chip (SoC) context to choose the most critical process knobs since most of processing budget to scale a technology node is already consumed by increasing process steps due to multiple patterning. In this paper we will show that the device parasitics and the interconnect resistance are the most critical performance scaling barriers for technology nodes beyond 7 nm (N7). We will demonstrate the impact of process and design knobs enabling performance and power improvements for the N7 node as defined in ITRS 2015 edition while still continuing to scale the area to limit the cost.