Fault Tolerant Clustering Revisited

In discrete k-center and k-median clustering, we are given a set of points P in a metric space M, and the task is to output a set C \subseteq ? P, |C| = k, such that the cost of clustering P using C is as small as possible. For k-center, the cost is the furthest a point has to travel to its nearest center, whereas for k-median, the cost is the sum of all point to nearest center distances. In the fault-tolerant versions of these problems, we are given an additional parameter 1 ?\leq \ell \leq ? k, such that when computing the cost of clustering, points are assigned to their \ell-th nearest-neighbor in C, instead of their nearest neighbor. We provide constant factor approximation algorithms for these problems that are both conceptually simple and highly practical from an implementation stand-point

Approximate Nearest Neighbor Search for Low Dimensional Queries

We study the Approximate Nearest Neighbor problem for metric spaces where the query points are constrained to lie on a subspace of low doubling dimension, while the data is high-dimensional. We show that this problem can be solved efficiently despite the high dimensionality of the data.Comment: 25 page

Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space

For a set of $n$ points in $\Re^d$, and parameters $k$ and \eps, we present a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time. Surprisingly, the space used by the data-structure is \Otilde (n /k); that is, the space used is sublinear in the input size if $k$ is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this, we provide a sublinear space data-structure that can estimate the density of a point set under various measures, including: \begin{inparaenum}[(i)] \item sum of distances of $k$ closest points to the query point, and \item sum of squared distances of $k$ closest points to the query point. \end{inparaenum} Our approach generalizes to other distance based estimation of densities of similar flavor. We also study the problem of approximating some of these quantities when using sampling. In particular, we show that a sample of size \Otilde (n /k) is sufficient, in some restricted cases, to estimate the above quantities. Remarkably, the sample size has only linear dependency on the dimension

Robust Proximity Search for Balls using Sublinear Space

Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm \epsilon)-approximate kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and \epsilon are provided at query time. If k and \epsilon are provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large

Purity based continuity bounds for quantum information measures

In quantum information theory, communication capacities are mostly given in terms of entropic formulas. Continuity of such entropic quantities are significant, as they lend themselves to maintain uniformity against perturbations of quantum states. Traditionally, continuity bounds have been provided in terms of the trace distance, which is a bonafide metric on the set of quantum states. In the present contribution we derive continuity bounds for various information measures based on the difference in purity of the concerned quantum states. In a finite-dimensional system, we establish continuity bounds for von Neumann entropy which depend only on purity distance and dimension of the system. We then obtain uniform continuity bounds for conditional von Neumann entropy in terms of purity distance which is free of the dimension of the conditioning subsystem. Furthermore, we derive the uniform continuity bounds for other entropic quantities like relative entropy distance, quantum mutual information and quantum conditional mutual information. As an application, we investigate the variation in squashed entanglement with respect to purity. We also obtain a bound to the quantum conditional mutual information of a quantum state which is arbitrarily close to a quantum Markov chain.Comment: We request suggestions and comment

Quantum conditional entropies and steerability of states with maximally mixed marginals

Quantum steering is an asymmetric correlation which occupies a place between entanglement and Bell nonlocality. In the paradigmatic scenario involving the protagonists Alice and Bob, the entangled state shared between them, is said to be steerable from Alice to Bob, if the steering assemblage on Bob's side do not admit a local hidden state (LHS) description. Quantum conditional entropies, on the other hand provide for another characterization of quantum correlations. Contrary to our common intuition conditional entropies for some entangled states can be negative, marking a significant departure from the classical realm. Quantum steering and quantum nonlocality in general, share an intricate relation with quantum conditional entropies. In the present contribution, we investigate this relationship. For a significant class, namely the two-qubit Weyl states we show that negativity of conditional R\'enyi 2-entropy and conditional Tsallis 2-entropy is a necessary and sufficient condition for the violation of a suitably chosen three settings steering inequality. With respect to the same inequality, we find an upper bound for the conditional R\'enyi 2-entropy, such that the general two-qubit state is steerable. Moving from a particular steering inequality to local hidden state descriptions, we show that some two-qubit Weyl states which admit a LHS model possess non-negative conditional R\'enyi 2-entropy. However, the same does not hold true for some non-Weyl states. Our study further investigates the relation between non-negativity of conditional entropy and LHS models in two-qudits for the isotropic and Werner states. There we find that whenever these states admit a LHS model, they possess a non-negative conditional R\'enyi 2-entropy. We then observe that the same holds true for a noisy variant of the two-qudit Werner state.Comment: 10 page

Approximating Minimization Diagrams and Generalized Proximity Search

We investigate the classes of functions whose minimization diagrams can be approximated efficiently in \Re^d. We present a general framework and a data-structure that can be used to approximate the minimization diagram of such functions. The resulting data-structure has near linear size and can answer queries in logarithmic time. Applications include approximating the Voronoi diagram of (additively or multiplicatively) weighted points. Our technique also works for more general distance functions, such as metrics induced by convex bodies, and the nearest furthest-neighbor distance to a set of point sets. Interestingly, our framework works also for distance functions that do not comply with the triangle inequality. For many of these functions no near-linear size approximation was known before

Space Exploration via Proximity Search

We investigate what computational tasks can be performed on a point set in $\Re^d$, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following: (A) One can compute an approximate bi-criteria $k$-center clustering of the point set, and more generally compute a greedy permutation of the point set. (B) One can decide if a query point is (approximately) inside the convex-hull of the point set. We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set

Efficient Algorithms for k-Regret Minimizing Sets

A regret minimizing set Q is a small size representation of a much larger database P so that user queries executed on Q return answers whose scores are not much worse than those on the full dataset. In particular, a k-regret minimizing set has the property that the regret ratio between the score of the top-1 item in Q and the score of the top-k item in P is minimized, where the score of an item is the inner product of the item\u27s attributes with a user\u27s weight (preference) vector. The problem is challenging because we want to find a single representative set Q whose regret ratio is small with respect to all possible user weight vectors. We show that k-regret minimization is NP-Complete for all dimensions d>=3, settling an open problem from Chester et al. [VLDB 2014]. Our main algorithmic contributions are two approximation algorithms, both with provable guarantees, one based on coresets and another based on hitting sets. We perform extensive experimental evaluation of our algorithms, using both real-world and synthetic data, and compare their performance against the solution proposed in [VLDB 14]. The results show that our algorithms are significantly faster and scalable to much larger sets than the greedy algorithm of Chester et al. for comparable quality answers