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Consider an instance of Euclidean kmeans or kmedians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1+ε) under a projection onto a random O(log(k /ε) / ε2)dimensional subspace. Further, the cost of every clustering is preserved within (1+ε). More generally, our result applies to any dimension reduction map satisfying a mild subGaussiantail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean kclustering with the distances raised to the pth power for any constant p. For kmeans, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for kmedians, it answers a question raised by Kannan.more » « less

Why are classifiers in high dimension vulnerable to “adversarial” perturbations? We show that it is likely not due to information theoretic limitations, but rather it could be due to computational constraints. First we prove that, for a broad set of classification tasks, the mere existence of a robust classifier implies that it can be found by a possibly exponentialtime algorithm with relatively few training examples. Then we give two particular classification tasks where learning a robust classifier is computationally intractable. More precisely we construct two binary classifications task in high dimensional space which are (i) information theoretically easy to learn robustly for large perturbations, (ii) efficiently learnable (nonrobustly) by a simple linear separator, (iii) yet are not efficiently robustly learnable, even for small perturbations. Specifically, for the first task hardness holds for any efficient algorithm in the statistical query (SQ) model, while for the second task we rule out any efficient algorithm under a cryptographic assumption. These examples give an exponential separation between classical learning and robust learning in the statistical query model or under a cryptographic assumption. It suggests that adversarial examples may be an unavoidable byproduct of computational limitations of learning algorithms.more » « less

We introduce and study the notion of *an outer biLipschitz extension* of a map between Euclidean spaces. The notion is a natural analogue of the notion of *a Lipschitz extension* of a Lipschitz map. We show that for every map f there exists an outer biLipschitz extension f′ whose distortion is greater than that of f by at most a constant factor. This result can be seen as a counterpart of the classic Kirszbraun theorem for outer biLipschitz extensions. We also study outer biLipschitz extensions of nearisometric maps and show upper and lower bounds for them. Then, we present applications of our results to prioritized and terminal dimension reduction problems, described next. We prove a *prioritized* variant of the Johnson–Lindenstrauss lemma: given a set of points X⊂ ℝd of size N and a permutation (”priority ranking”) of X, there exists an embedding f of X into ℝO(logN) with distortion O(loglogN) such that the point of rank j has only O(log3 + ε j) nonzero coordinates – more specifically, all but the first O(log3+ε j) coordinates are equal to 0; the distortion of f restricted to the first j points (according to the ranking) is at most O(loglogj). The result makes a progress towards answering an open question by Elkin, Filtser, and Neiman about prioritized dimension reductions. We prove that given a set X of N points in ℜd, there exists a *terminal* dimension reduction embedding of ℝd into ℝd′, where d′ = O(logN/ε4), which preserves distances x−y between points x∈ X and y ∈ ℝd, up to a multiplicative factor of 1 ± ε. This improves a recent result by Elkin, Filtser, and Neiman. The dimension reductions that we obtain are nonlinear, and this nonlinearity is necessary.more » « less

We introduce a new distancepreserving compact representation of multidimensional pointsets. Given n points in a ddimensional space where each coordinate is represented using B bits (i.e., dB bits per point), it produces a representation of size O( d log(d B/epsilon) +log n) bits per point from which one can approximate the distances up to a factor of 1 + epsilon. Our algorithm almost matches the recent bound of Indyk et al, 2017} while being much simpler. We compare our algorithm to Product Quantization (PQ) (Jegou et al, 2011) a state of the art heuristic metric compression method. We evaluate both algorithms on several data sets: SIFT, MNIST, New York City taxi time series and a synthetic onedimensional data set embedded in a highdimensional space. Our algorithm produces representations that are comparable to or better than those produced by PQ, while having provable guarantees on its performance.more » « less