arXiv subjects
Publications and source records attributed to John Wright.
The k-means problem consists of finding k centers in the d-dimensional Euclidean space that minimize the sum of the squared distances of all points in an input set P to their closest respective center. Awasthi et. al. recently showed that there exists a constant c > 1 such that it is NP-hard to approximate the k-means objective within a factor of c. We establish that the constant c is at least 1.0013.
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.
We propose a batchwise monotone algorithm for dictionary learning. Unlike the state-of-the-art dictionary learning algorithms which impose sparsity constraints on a sample-by-sample basis, we instead treat the samples as a batch, and impose the sparsity constraint on the whole. The benefit of batchwise optimization is that the non-zeros can be better allocated across the samples, leading to a better approximation of the whole. To accomplish this, we propose procedures to switch non-zeros in both rows and columns in the support of the coefficient matrix to reduce the reconstruction error. We prove in the proposed support switching procedure the objective of the algorithm, i.e., the reconstruction error, decreases monotonically and converges. Furthermore, we introduce a block orthogonal matching pursuit algorithm that also operates on sample batches to provide a warm start. Experiments on both natural image patches and UCI data sets show that the proposed algorithm produces a better approximation with the same sparsity levels compared to the state-of-the-art algorithms.
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.
We report on progress towards implementing mixed ion species quantum information processing for a scalable ion trap architecture. Mixed species chains may help solve several problems with scaling ion trap quantum computation to large numbers of qubits. Initial temperature measurements of linear Coulomb crystals containing barium and ytterbium ions indicate that the mass difference does not significantly impede cooling at low ion numbers. Average motional occupation numbers are estimated to be $\bar{n} \approx 130$ quanta per mode for chains with small numbers of ions, which is within a factor of three of the Doppler limit for barium ions in our trap. We also discuss generation of ion-photon entanglement with barium ions with a fidelity of $F \ge 0.84$, which is an initial step towards remote ion-ion coupling in a more scalable quantum information architecture. Further, we are working to implement these techniques in surface traps in order to exercise greater control over ion chain ordering and positioning.
The absolute-magnitude distributions of seven supernova types are presented. The data used here were primarily taken from the Asiago Supernova Catalogue, but were supplemented with additional data. We accounted for both foreground and host-galaxy extinction. A bootstrap method is used to correct the samples for Malmquist bias. Separately, we generate volume-limited samples, restricted to events within 100 Mpc. We find that the superluminous events (M_B < -21) make up only about 0.1% of all supernovae in the bias-corrected sample. The subluminous events (M_B > -15) make up about 3%. The normal Ia distribution was the brightest with a mean absolute blue magnitude of -19.25. The IIP distribution was the dimmest at -16.75.
Motivated by vision tasks such as robust face and object recognition, we consider the following general problem: given a collection of low-dimensional linear subspaces in a high-dimensional ambient (image) space, and a query point (image), efficiently determine the nearest subspace to the query in $\ell^1$ distance. In contrast to the naive exhaustive search which entails large-scale linear programs, we show that the computational burden can be cut down significantly by a simple two-stage algorithm: (1) projecting the query and data-base subspaces into lower-dimensional space by random Cauchy matrix, and solving small-scale distance evaluations (linear programs) in the projection space to locate candidate nearest; (2) with few candidates upon independent repetition of (1), getting back to the high-dimensional space and performing exhaustive search. To preserve the identity of the nearest subspace with nontrivial probability, the projection dimension typically is low-order polynomial of the subspace dimension multiplied by logarithm of number of the subspaces (Theorem 2.1). The reduced dimensionality and hence complexity renders the proposed algorithm particularly relevant to vision application such as robust face and object instance recognition that we investigate empirically.
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.
We have developed a vacuum chamber and control system for rapid testing of microfabricated surface ion traps. Our system is modular in design and is based on an in-vacuum printed circuit board with integrated filters. We have used this system to successfully trap and cool barium ions and have achieved ion 'dark' lifetimes of 31.6 s +- 3.4 s with controlled shuttling of ions. We provide a detailed description of the ion trap system including the in-vacuum materials used, control electronics and neutral atom source. We discuss the challenges presented in achieving a system which can work reliably over two years of operations in which the trap under test was changed at least 10 times.
Given $f:\{-1, 1\}^n \rightarrow \{-1, 1\}$, define the \emph{spectral distribution} of $f$ to be the distribution on subsets of $[n]$ in which the set $S$ is sampled with probability $\widehat{f}(S)^2$. Then the Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai (1996) states that there is some absolute constant $C$ such that $\operatorname{H}[\widehat{f}^2] \leq C\cdot\operatorname{Inf}[f]$. Here, $\operatorname{H}[\widehat{f}^2]$ denotes the Shannon entropy of $f$'s spectral distribution, and $\operatorname{Inf}[f]$ is the total influence of $f$. This conjecture is one of the major open problems in the analysis of Boolean functions, and settling it would have several interesting consequences. Previous results on the FEI conjecture have been largely through direct calculation. In this paper we study a natural interpretation of the conjecture, which states that there exists a communication protocol which, given subset $S$ of $[n]$ distributed as $\widehat{f}^2$, can communicate the value of $S$ using at most $C\cdot\operatorname{Inf}[f]$ bits in expectation. Using this interpretation, we are able show the following results: 1. First, if $f$ is computable by a read-$k$ decision tree, then $\operatorname{H}[\widehat{f}^2] \leq 9k\cdot \operatorname{Inf}[f]$. 2. Next, if $f$ has $\operatorname{Inf}[f] \geq 1$ and is computable by a decision tree with expected depth $d$, then $\operatorname{H}[\widehat{f}^2] \leq 12d\cdot \operatorname{Inf}[f]$. 3. Finally, we give a new proof of the main theorem of O'Donnell and Tan (ICALP 2013), i.e. that their FEI$^+$ conjecture composes. In addition, we show that natural improvements to our decision tree results would be sufficient to prove the FEI conjecture in its entirety. We believe that our methods give more illuminating proofs than previous results about the FEI conjecture.
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}.
Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a $K$-way tensor of length $n$ and Tucker rank $r$ from Gaussian measurements requires $Ω(r n^{K-1})$ observations. In contrast, a certain (intractable) nonconvex formulation needs only $O(r^K + nrK)$ observations. We introduce a very simple, new convex relaxation, which partially bridges this gap. Our new formulation succeeds with $O(r^{\lfloor K/2 \rfloor}n^{\lceil K/2 \rceil})$ observations. While these results pertain to Gaussian measurements, simulations strongly suggest that the new norm also outperforms the sum of nuclear norms for tensor completion from a random subset of entries. Our lower bound for the sum-of-nuclear-norms model follows from a new result on recovering signals with multiple sparse structures (e.g. sparse, low rank), which perhaps surprisingly demonstrates the significant suboptimality of the commonly used recovery approach via minimizing the sum of individual sparsity inducing norms (e.g. $l_1$, nuclear norm). Our new formulation for low-rank tensor recovery however opens the possibility in reducing the sample complexity by exploiting several structures jointly.
Illumination variation remains a central challenge in object detection and recognition. Existing analyses of illumination variation typically pertain to convex, Lambertian objects, and guarantee quality of approximation in an average case sense. We show that it is possible to build V(vertex)-description convex cone models with worst-case performance guarantees, for non-convex Lambertian objects. Namely, a natural verification test based on the angle to the constructed cone guarantees to accept any image which is sufficiently well-approximated by an image of the object under some admissible lighting condition, and guarantees to reject any image that does not have a sufficiently good approximation. The cone models are generated by sampling point illuminations with sufficient density, which follows from a new perturbation bound for point images in the Lambertian model. As the number of point images required for guaranteed verification may be large, we introduce a new formulation for cone preserving dimensionality reduction, which leverages tools from sparse and low-rank decomposition to reduce the complexity, while controlling the approximation error with respect to the original cone.
Motivated by vision tasks such as robust face and object recognition, we consider the following general problem: given a collection of low-dimensional linear subspaces in a high-dimensional ambient (image) space and a query point (image), efficiently determine the nearest subspace to the query in $\ell^1$ distance. We show in theory that Cauchy random embedding of the objects into significantly-lower-dimensional spaces helps preserve the identity of the nearest subspace with constant probability. This offers the possibility of efficiently selecting several candidates for accurate search. We sketch preliminary experiments on robust face and digit recognition to corroborate our theory.
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.
We consider the problem of learning sparsely used dictionaries with an arbitrary square dictionary and a random, sparse coefficient matrix. We prove that $O (n \log n)$ samples are sufficient to uniquely determine the coefficient matrix. Based on this proof, we design a polynomial-time algorithm, called Exact Recovery of Sparsely-Used Dictionaries (ER-SpUD), and prove that it probably recovers the dictionary and coefficient matrix when the coefficient matrix is sufficiently sparse. Simulation results show that ER-SpUD reveals the true dictionary as well as the coefficients with probability higher than many state-of-the-art algorithms.
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.
In this paper, we study the problem of decomposing a superposition of a low-rank matrix and a sparse matrix when a relatively few linear measurements are available. This problem arises in many data processing tasks such as aligning multiple images or rectifying regular texture, where the goal is to recover a low-rank matrix with a large fraction of corrupted entries in the presence of nonlinear domain transformation. We consider a natural convex heuristic to this problem which is a variant to the recently proposed Principal Component Pursuit. We prove that under suitable conditions, this convex program guarantees to recover the correct low-rank and sparse components despite reduced measurements. Our analysis covers both random and deterministic measurement models.