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Ryan O'Donnell

Publications and source records attributed to Ryan O'Donnell.

At least 73 records · Page 4Linked to original sources
arXiv2016

Hypercontractive inequalities via SOS, and the Frankl--Rödl graph

Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant $0 < γ\leq 1/4$, the SOS/Lasserre SDP hierarchy at degree $4\lceil \frac{1}{4γ}\rceil$ certifies the statement "the maximum independent set in the Frankl--Rödl graph $\mathrm{FR}^{n}_γ$ has fractional size~$o(1)$". Here $\mathrm{FR}^{n}_γ = (V,E)$ is the graph with $V = \{0,1\}^n$ and $(x,y) \in E$ whenever $Δ(x,y) = (1-γ)n$ (an even integer). In particular, we show the degree-$4$ SOS algorithm certifies the chromatic number lower bound "$χ(\mathrm{FR}^{n}_{1/4}) = ω(1)$", even though $\mathrm{FR}^{n}_{1/4}$ is the canonical integrality gap instance for which standard SDP relaxations cannot even certify "$χ(\mathrm{FR}^{n}_{1/4}) > 3$". Finally, we also give an SOS proof of (a generalization of) the sharp $(2,q)$-hypercontractive inequality for any even integer $q$.

cs.CC↗
arXiv2016

Remarks on the Most Informative Function Conjecture at fixed mean

In 2013, Courtade and Kumar posed the following problem: Let $\boldsymbol{x} \sim \{\pm 1\}^n$ be uniformly random, and form $\boldsymbol{y} \sim \{\pm 1\}^n$ by negating each bit of $\boldsymbol{x}$ independently with probability $α$. Is it true that the mutual information $I(f(\boldsymbol{x}) \mathbin{;} \boldsymbol{y})$ is maximized among $f:\{\pm 1\}^n \to \{\pm 1\}$ by $f(x) = x_1$? We do not resolve this problem. Instead, we make a couple of observations about the fixed-mean version of the conjecture. We show that Courtade and Kumar's stronger Lex Conjecture fails for small noise rates. We also prove a continuous version of the conjecture on the sphere and show that it implies the previously-known analogue for Gaussian space.

cs.IT↗
arXiv2015

Polynomial bounds for decoupling, with applications

Let f(x) = f(x_1, ..., x_n) = \sum_{|S| <= k} a_S \prod_{i \in S} x_i be an n-variate real multilinear polynomial of degree at most k, where S \subseteq [n] = {1, 2, ..., n}. For its "one-block decoupled" version, f~(y,z) = \sum_{|S| <= k} a_S \sum_{i \in S} y_i \prod_{j \in Sı} z_j, we show tail-bound comparisons of the form Pr[|f~(y,z)| > C_k t] <= D_k Pr[f(x) > t]. Our constants C_k, D_k are significantly better than those known for "full decoupling". For example, when x, y, z are independent Gaussians we obtain C_k = D_k = O(k); when x, y, z, Rademacher random variables we obtain C_k = O(k^2), D_k = k^{O(k)}. By contrast, for full decoupling only C_k = D_k = k^{O(k)} is known in these settings. We describe consequences of these results for query complexity (related to conjectures of Aaronson and Ambainis) and for analysis of Boolean functions (including an optimal sharpening of the DFKO Inequality).

cs.DM↗
arXiv2015

Efficient quantum tomography

In the quantum state tomography problem, one wishes to estimate an unknown $d$-dimensional mixed quantum state $ρ$, given few copies. We show that $O(d/ε)$ copies suffice to obtain an estimate $\hatρ$ that satisfies $\|\hatρ - ρ\|_F^2 \leq ε$ (with high probability). An immediate consequence is that $O(\mathrm{rank}(ρ) \cdot d/ε^2) \leq O(d^2/ε^2)$ copies suffice to obtain an $ε$-accurate estimate in the standard trace distance. This improves on the best known prior result of $O(d^3/ε^2)$ copies for full tomography, and even on the best known prior result of $O(d^2\log(d/ε)/ε^2)$ copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension. Next, we generalize these results to show that one can perform efficient principal component analysis on $ρ$. Our main result is that $O(k d/ε^2)$ copies suffice to output a rank-$k$ approximation $\hatρ$ whose trace distance error is at most $ε$ more than that of the best rank-$k$ approximator to $ρ$. This subsumes our above trace distance tomography result and generalizes it to the case when $ρ$ is not guaranteed to be of low rank. A key part of the proof is the analogous generalization of our spectrum-learning results: we show that the largest $k$ eigenvalues of $ρ$ can be estimated to trace-distance error $ε$ using $O(k^2/ε^2)$ copies. In turn, this result relies on a new coupling theorem concerning the Robinson-Schensted-Knuth algorithm that should be of independent combinatorial interest.

quant-ph↗
arXiv2015

Beating the random assignment on constraint satisfaction problems of bounded degree

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + Ω(1/\sqrt{D})$ fraction of constraints, where $D$ is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a \emph{quantum} algorithm to find an assignment satisfying a $\frac{1}{2} + Ω(D^{-3/4})$ fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a $μ+ Ω(1/\sqrt{D})$ fraction of constraints, where $μ$ is the fraction that would be satisfied by a uniformly random assignment.

cs.CC↗
arXiv2015

How to refute a random CSP

Let $P$ be a $k$-ary predicate over a finite alphabet. Consider a random CSP$(P)$ instance $I$ over $n$ variables with $m$ constraints. When $m \gg n$ the instance $I$ will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When $P$ is the $3$-ary OR predicate, this is the well studied problem of refuting random $3$-SAT formulas, and an efficient algorithm is known only when $m \gg n^{3/2}$. Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a $k$-ary predicate, having $m \gg n^{\lceil k/2 \rceil}$ constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if $P$ fails to support a $t$-wise uniform distribution, then there is an efficient algorithm that refutes random CSP$(P)$ instances $I$ whp when $m \gg n^{t/2}$. Indeed, our algorithm will "somewhat strongly" refute $I$, certifying $\mathrm{Opt}(I) \leq 1-Ω_k(1)$, if $t = k$ then we get the strongest possible refutation, certifying $\mathrm{Opt}(I) \leq \mathrm{E}[P] + o(1)$. This last result is new even in the context of random $k$-SAT. Regarding the optimality of our $m \gg n^{t/2}$ requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP$(P)$ may be impossible when $m \ll n^{t/2}$. Thus there is an indication our algorithm's dependence on $m$ is optimal for every $P$, at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the $O(1)$-round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz.

cs.CC↗
arXiv2015

Quantum Spectrum Testing

In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given $n$ copies of a mixed state $ρ\in\mathbb{C}^{d\times d}$ and the goal is to distinguish whether $ρ$'s spectrum satisfies some property $\mathcal{P}$ or is at least $ε$-far in $\ell_1$-distance from satisfying $\mathcal{P}$. This problem was promoted in the survey of Montanaro and de Wolf under the name of testing unitarily invariant properties of mixed states. It is the natural quantum analogue of the classical problem of testing symmetric properties of probability distributions. Here, the hope is for algorithms with subquadratic copy complexity in the dimension $d$. This is because the "empirical Young diagram (EYD) algorithm" can estimate the spectrum of a mixed state up to $ε$-accuracy using only $\widetilde{O}(d^2/ε^2)$ copies. In this work, we show that given a mixed state $ρ\in\mathbb{C}^{d\times d}$: (i) $Θ(d/ε^2)$ copies are necessary and sufficient to test whether $ρ$ is the maximally mixed state, i.e., has spectrum $(\frac1d, ..., \frac1d)$; (ii) $Θ(r^2/ε)$ copies are necessary and sufficient to test with one-sided error whether $ρ$ has rank $r$, i.e., has at most $r$ nonzero eigenvalues; (iii) $\widetildeΘ(r^2/Δ)$ copies are necessary and sufficient to distinguish whether $ρ$ is maximally mixed on an $r$-dimensional or an $(r+Δ)$-dimensional subspace; and (iv) The EYD algorithm requires $Ω(d^2/ε^2)$ copies to estimate the spectrum of $ρ$ up to $ε$-accuracy, nearly matching the known upper bound. In addition, we simplify part of the proof of the upper bound. Our techniques involve the asymptotic representation theory of the symmetric group; in particular Kerov's algebra of polynomial functions on Young diagrams.

quant-ph↗
arXiv2014

Social choice, computational complexity, Gaussian geometry, and Boolean functions

We describe a web of connections between the following topics: the mathematical theory of voting and social choice; the computational complexity of the Maximum Cut problem; the Gaussian Isoperimetric Inequality and Borell's generalization thereof; the Hypercontractive Inequality of Bonami; and, the analysis of Boolean functions. A major theme is the technique of reducing inequalities about Gaussian functions to inequalities about Boolean functions f : {-1,1}^n -> {-1,1}, and then using induction on n to further reduce to inequalities about functions f : {-1,1} -> {-1,1}. We especially highlight De, Mossel, and Neeman's recent use of this technique to prove the Majority Is Stablest Theorem and Borell's Isoperimetric Inequality simultaneously.

math.PR↗
arXiv2014

Conditioning and covariance on caterpillars

Let $X_1, \dots, X_n$ be joint $\{ \pm 1\}$-valued random variables. It is known that conditioning on a random subset of $O(1/ε^2)$ of them reduces their average pairwise covariance to below $ε$ (in expectation). We conjecture that $O(1/ε^2)$ can be improved to $O(1/ε)$. The motivation for the problem and our conjectured improvement comes from the theory of global correlation rounding for convex relaxation hierarchies. We suggest attempting the conjecture in the case that $X_1, \dots, X_n$ are the leaves of an information flow tree. We prove the conjecture in the case that the information flow tree is a caterpillar graph (similar to a two-state hidden Markov model).

cs.IT↗
arXiv2014

Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs

In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. On infinite unimodular transitive graphs (e.g., Cayley graphs), when initial opinions are chosen from a distribution that is invariant with respect to the graph automorphism group, we show that the opinion of each agent almost surely either converges, or else eventually oscillates with period two; this is known to hold for finite graphs, but not for all infinite graphs. On Erdős-Rényi random graphs with degrees $Ω(\sqrt{n})$, we show that when initial opinions are chosen i.i.d. then agents all converge to the initial majority opinion, with constant probability. Conversely, on random 4-regular finite graphs, we show that with high probability different agents converge to different opinions.

math.PR↗
arXiv2014

Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs

Building on work of Cai, Fürer, and Immerman \cite{CFI92}, we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic $n$-vertex graphs $G$ and $H$ such that any sum-of-squares (SOS) proof of nonisomorphism requires degree $Ω(n)$. In other words, we show an $Ω(n)$-round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs $G$ and $H$ which are not even $(1-10^{-14})$-isomorphic. (Here we say that two $n$-vertex, $m$-edge graphs $G$ and $H$ are $α$-isomorphic if there is a bijection between their vertices which preserves at least $αm$ edges.) Our second result is that under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust} Graph Isomorphism problem is hard. I.e.\ for every $ε> 0$, there is no efficient algorithm which can distinguish graph pairs which are $(1-ε)$-isomorphic from pairs which are not even $(1-ε_0)$-isomorphic for some universal constant $ε_0$. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest.

cs.CC↗
arXiv2013

A composition theorem for parity kill number

In this work, we study the parity complexity measures ${\mathsf{C}^{\oplus}_{\min}}[f]$ and ${\mathsf{DT^{\oplus}}}[f]$. ${\mathsf{C}^{\oplus}_{\min}}[f]$ is the \emph{parity kill number} of $f$, the fewest number of parities on the input variables one has to fix in order to "kill" $f$, i.e. to make it constant. ${\mathsf{DT^{\oplus}}}[f]$ is the depth of the shortest \emph{parity decision tree} which computes $f$. These complexity measures have in recent years become increasingly important in the fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and pseudorandomness \cite{BK12, Sha11, CT13}. Our main result is a composition theorem for ${\mathsf{C}^{\oplus}_{\min}}$. The $k$-th power of $f$, denoted $f^{\circ k}$, is the function which results from composing $f$ with itself $k$ times. We prove that if $f$ is not a parity function, then ${\mathsf{C}^{\oplus}_{\min}}[f^{\circ k}] \geq Ω({\mathsf{C}_{\min}}[f]^{k}).$ In other words, the parity kill number of $f$ is essentially supermultiplicative in the \emph{normal} kill number of $f$ (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity complexity measures of $\mathsf{Sort}^{\circ k}$ and $\mathsf{HI}^{\circ k}$. Here $\mathsf{Sort}$ is the sort function due to Ambainis \cite{Amb06}, and $\mathsf{HI}$ is Kushilevitz's hemi-icosahedron function \cite{NW95}. In doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and \cite{TWXZ13}.

cs.CC↗
arXiv2013

Markov chain methods for small-set expansion

Consider a finite irreducible Markov chain with invariant distribution $π$. We use the inner product induced by $π$ and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set expansion problem. For example, Steurer showed a tight connection between the number of small eigenvalues of a graph's Laplacian and the expansion of small sets in that graph. We give a simplified proof which generalizes to the nonregular, directed case. This result implies an approximation algorithm for an "analytic" version of the Small-Set Expansion Problem, which, in turn, immediately gives an approximation algorithm for Small-Set Expansion. We also give a simpler proof of a lower bound on the probability that a random walk stays within a set; this result was used in some recent works on finding small sparse cuts.

cs.DS↗
arXiv2013

A composition theorem for the Fourier Entropy-Influence conjecture

The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96] seeks to relate two fundamental measures of Boolean function complexity: it states that $H[f] \leq C Inf[f]$ holds for every Boolean function $f$, where $H[f]$ denotes the spectral entropy of $f$, $Inf[f]$ is its total influence, and $C > 0$ is a universal constant. Despite significant interest in the conjecture it has only been shown to hold for a few classes of Boolean functions. Our main result is a composition theorem for the FEI conjecture. We show that if $g_1,...,g_k$ are functions over disjoint sets of variables satisfying the conjecture, and if the Fourier transform of $F$ taken with respect to the product distribution with biases $E[g_1],...,E[g_k]$ satisfies the conjecture, then their composition $F(g_1(x^1),...,g_k(x^k))$ satisfies the conjecture. As an application we show that the FEI conjecture holds for read-once formulas over arbitrary gates of bounded arity, extending a recent result [OWZ11] which proved it for read-once decision trees. Our techniques also yield an explicit function with the largest known ratio of $C \geq 6.278$ between $H[f]$ and $Inf[f]$, improving on the previous lower bound of 4.615.

cs.CC↗
arXiv2012

Approximability and proof complexity

This work is concerned with the proof-complexity of certifying that optimization problems do \emph{not} have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any $n$-variable degree-$d$ proof can be found in time $n^{O(d)}$. Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "$d/2$-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-$d$ SOS proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al.\ can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 ($> .878$) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?

cs.CC↗
arXiv2012

New NP-hardness results for 3-Coloring and 2-to-1 Label Cover

We show that given a 3-colorable graph, it is NP-hard to find a 3-coloring with $(16/17 + \eps)$ of the edges bichromatic. In a related result, we show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a $(23/24 + \eps)$-satisfying assignment.

cs.CC↗