82 research outputs found
K-median clustering, model-based compressive sensing, and sparse recovery for earth mover distance
We initiate the study of sparse recovery problems under the Earth-Mover Distance (EMD). Specifically, we design a distribution over m x n matrices A such that for any x, given Ax, we can recover a k-sparse approximation to x under the EMD distance. One construction yields m=O(k log (n/k)) and a 1 + ε approximation factor, which matches the best achievable bound for other error measures, such as the l[subscript 1] norm.
Our algorithms are obtained by exploiting novel connections to other problems and areas, such as streaming algorithms for k-median clustering and model-based compressive sensing. We also provide novel algorithms and results for the latter problems
Differentially Private Heatmaps
We consider the task of producing heatmaps from users' aggregated data while
protecting their privacy. We give a differentially private (DP) algorithm for
this task and demonstrate its advantages over previous algorithms on real-world
datasets.
Our core algorithmic primitive is a DP procedure that takes in a set of
distributions and produces an output that is close in Earth Mover's Distance to
the average of the inputs. We prove theoretical bounds on the error of our
algorithm under a certain sparsity assumption and that these are near-optimal.Comment: To appear in AAAI 202
On Model-Based RIP-1 Matrices
The Restricted Isometry Property (RIP) is a fundamental property of a matrix
enabling sparse recovery. Informally, an m x n matrix satisfies RIP of order k
in the l_p norm if ||Ax||_p \approx ||x||_p for any vector x that is k-sparse,
i.e., that has at most k non-zeros. The minimal number of rows m necessary for
the property to hold has been extensively investigated, and tight bounds are
known. Motivated by signal processing models, a recent work of Baraniuk et al
has generalized this notion to the case where the support of x must belong to a
given model, i.e., a given family of supports. This more general notion is much
less understood, especially for norms other than l_2. In this paper we present
tight bounds for the model-based RIP property in the l_1 norm. Our bounds hold
for the two most frequently investigated models: tree-sparsity and
block-sparsity. We also show implications of our results to sparse recovery
problems.Comment: Version 3 corrects a few errors present in the earlier version. In
particular, it states and proves correct upper and lower bounds for the
number of rows in RIP-1 matrices for the block-sparse model. The bounds are
of the form k log_b n, not k log_k n as stated in the earlier versio
Model-based compressive sensing with Earth Mover's Distance constraints
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 71-72).In compressive sensing, we want to recover ... from linear measurements of the form ... describes the measurement process. Standard results in compressive sensing show that it is possible to exactly recover the signal x from only m ... measurements for certain types of matrices. Model-based compressive sensing reduces the number of measurements even further by limiting the supports of x to a subset of the ... possible supports. Such a family of supports is called a structured sparsity model. In this thesis, we introduce a structured sparsity model for two-dimensional signals that have similar support in neighboring columns. We quantify the change in support between neighboring columns with the Earth Mover's Distance (EMD), which measures both how many elements of the support change and how far the supported elements move. We prove that for a reasonable limit on the EMD between adjacent columns, we can recover signals in our model from only ... measurements, where w is the width of the signal. This is an asymptotic improvement over the ... bound in standard compressive sensing. While developing the algorithmic tools for our proposed structured sparsity model, we also extend the model-based compressed sensing framework. In order to use a structured sparsity model in compressive sensing, we need a model projection algorithm that, given an arbitrary signal x, returns the best approximation in the model. We relax this constraint and develop a variant of IHT, an existing sparse recovery algorithm, that works with approximate model projection algorithms.by Ludwig Schmidt.S.M
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