Inferring Rankings Using Constrained Sensing

We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over $n$ elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size $\ll$ domain size). Our recovery method is based on finding the sparsest solution (through $\ell_0$ optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through $\ell_0$ optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as $n \to \infty$. $\ell_0$ optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, $\ell_1$ optimization, to find the sparsest solution. However, we show that $\ell_1$ optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as $n \to \infty$. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible.Comment: 19 page

The conditional permutation test for independence while controlling for confounders

We propose a general new method, the conditional permutation test, for testing the conditional independence of variables $X$ and $Y$ given a potentially high-dimensional random vector $Z$ that may contain confounding factors. The proposed test permutes entries of $X$ non-uniformly, so as to respect the existing dependence between $X$ and $Z$ and thus account for the presence of these confounders. Like the conditional randomization test of Cand\`es et al. (2018), our test relies on the availability of an approximation to the distribution of $X \mid Z$. While Cand\`es et al. (2018)'s test uses this estimate to draw new $X$ values, for our test we use this approximation to design an appropriate non-uniform distribution on permutations of the $X$ values already seen in the true data. We provide an efficient Markov Chain Monte Carlo sampler for the implementation of our method, and establish bounds on the Type I error in terms of the error in the approximation of the conditional distribution of $X\mid Z$, finding that, for the worst case test statistic, the inflation in Type I error of the conditional permutation test is no larger than that of the conditional randomization test. We validate these theoretical results with experiments on simulated data and on the Capital Bikeshare data set.Comment: 31 pages, 4 figure

Symmetry implies independence

Given a quantum system consisting of many parts, we show that symmetry of the system's state, i.e., invariance under swappings of the subsystems, implies that almost all of its parts are virtually identical and independent of each other. This result generalises de Finetti's classical representation theorem for infinitely exchangeable sequences of random variables as well as its quantum-mechanical analogue. It has applications in various areas of physics as well as information theory and cryptography. For example, in experimental physics, one typically collects data by running a certain experiment many times, assuming that the individual runs are mutually independent. Our result can be used to justify this assumption.Comment: LaTeX, contains 4 figure

Performing Nonlinear Blind Source Separation with Signal Invariants

Given a time series of multicomponent measurements x(t), the usual objective of nonlinear blind source separation (BSS) is to find a "source" time series s(t), comprised of statistically independent combinations of the measured components. In this paper, the source time series is required to have a density function in (s,ds/dt)-space that is equal to the product of density functions of individual components. This formulation of the BSS problem has a solution that is unique, up to permutations and component-wise transformations. Separability is shown to impose constraints on certain locally invariant (scalar) functions of x, which are derived from local higher-order correlations of the data's velocity dx/dt. The data are separable if and only if they satisfy these constraints, and, if the constraints are satisfied, the sources can be explicitly constructed from the data. The method is illustrated by using it to separate two speech-like sounds recorded with a single microphone.Comment: 8 pages, 3 figure

IV models of ordered choice

This paper studies single equation instrumental variable models of ordered choice in which explanatory variables may be endogenous. The models are weakly restrictive, leaving unspecified the mechanism that generates endogenous variables. These incomplete models are set, not point, identifying for parametrically (e.g. ordered probit) or nonparametrically specified structural functions. The paper gives results on the properties of the identified set for the case in which potentially endogenous explanatory variables are discrete. The results are used as the basis for calculations showing the rate of shrinkage of identified sets as the number of classes in which the outcome is categorised increases

The Entropy of Backwards Analysis

Backwards analysis, first popularized by Seidel, is often the simplest most elegant way of analyzing a randomized algorithm. It applies to incremental algorithms where elements are added incrementally, following some random permutation, e.g., incremental Delauney triangulation of a pointset, where points are added one by one, and where we always maintain the Delauney triangulation of the points added thus far. For backwards analysis, we think of the permutation as generated backwards, implying that the $i$th point in the permutation is picked uniformly at random from the $i$ points not picked yet in the backwards direction. Backwards analysis has also been applied elegantly by Chan to the randomized linear time minimum spanning tree algorithm of Karger, Klein, and Tarjan. The question considered in this paper is how much randomness we need in order to trust the expected bounds obtained using backwards analysis, exactly and approximately. For the exact case, it turns out that a random permutation works if and only if it is minwise, that is, for any given subset, each element has the same chance of being first. Minwise permutations are known to have $\Theta(n)$ entropy, and this is then also what we need for exact backwards analysis. However, when it comes to approximation, the two concepts diverge dramatically. To get backwards analysis to hold within a factor $\alpha$, the random permutation needs entropy $\Omega(n/\alpha)$. This contrasts with minwise permutations, where it is known that a $1+\varepsilon$ approximation only needs $\Theta(\log (n/\varepsilon))$ entropy. Our negative result for backwards analysis essentially shows that it is as abstract as any analysis based on full randomness