Anup Bhattacharya

A Tight Lower Bound Instance for k-means++ in Constant Dimension

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial k centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the first center randomly from the given points. For i > 1, pick a point to be the i th center with probability proportional to the square of the Euclidean distance of this point to the closest previously (i − 1) chosen centers.

GPU-Based Implementation of 128-Bit Secure Eta Pairing over a Binary Field

Eta pairing on a supersingular elliptic curve over the binary field \(F_{2^{1223}}\) used to offer 128-bit security, and has been studied extensively for efficient implementations. In this paper, we report our GPU-based implementations of this algorithm on an NVIDIA Tesla C2050 platform. We propose efficient parallel implementation strategies for multiplication, square, square root and inverse in the underlying field. Our implementations achieve the best performance when Lopez-Dahab multiplication with four-bit precomputations is used in conjunction with one-level Karatsuba multiplication. We have been able to compute up to 566 eta pairings per second. To the best of our knowledge, ours is the fastest GPU-based implementation of eta pairing. It is about twice as fast as the only reported GPU implementation, and about five times as fast as the fastest reported single-core SIMD implementation. We estimate that the NVIDIA GTX 480 platform is capable of producing the fastest known software implementation of eta pairing.

Faster Algorithms for the Constrained k-means Problem

The classical center based clustering problems such as k-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. For instance, consider the r-gather clustering problem where there is an additional constraint that each of the clusters should have at least r points or the capacitated clustering problem where there is an upper bound on the cluster sizes. Consider a variant of the k-means problem that may be regarded as a general version of such problems. Here, the optimal clusters O1, ..., Ok are an arbitrary partition of the dataset and the goal is to output k-centers c1, ..., ck such that the objective function ∑i=1k∑x∈Oi||x−ci||2

Sampling in Space Restricted Settings

{\sum }_{i = 1}^{k} {\sum }_{x \in O_{i}} ||x - c_{i}||^{2}

Approximate Correlation Clustering Using Same-Cluster Queries

is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of k centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such k centers such that at least one of these k centers behaves well. Given an error parameter ε > 0, let ℓ denote the size of the smallest list of k-centers such that at least one of the k-centers gives a (1 + ε) approximation w.r.t. the objective function above. In this paper, we show an upper bound on ℓ by giving a randomized algorithm that outputs a list of 2Õ(k/ε)

Tight lower bound instances for k-means++ in two dimensions

2^{\tilde {O}(k/\varepsilon )}

Use of SIMD Features to Speed up Eta Pairing

k-centers. We also give a closely matching lower bound of 2Ω~(k/ε)

Approximate Clustering with Same-Cluster Queries

2^{\tilde {\Omega }(k/\sqrt {\varepsilon })}

Faster Algorithms for the Constrained k-Means Problem

. Moreover, our algorithm runs in time Ond⋅2Õ(k/ε)

Triangle Estimation using Polylogarithmic Queries.

O \left (n d \cdot 2^{\tilde {O}(k/\varepsilon )} \right )