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- Rafail Ostrovsky, Yuval Rabani, Leonard J. Schulman, Chaitanya Swamy
- 2006 47th Annual IEEE Symposium on Foundations of…
- 2006

We investigate variants of Lloyd's heuristic for clustering high-dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a <i>clusterability</i> criterion for data sets. We present variants of Lloyd's heuristic that… (More)

- Moni Naor, Leonard J. Schulman, Aravind Srinivasan
- FOCS
- 1995

We present a fairly general method for nding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2 k… (More)

- Oded Goldreich, Howard J. Karloff, Leonard J. Schulman, Luca Trevisan
- computational complexity
- 2001

We prove that if a linear error-correcting code C:{0, 1} n →{0, 1} m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2Ω (n). We also present several extensions of this result. We show a reduction from the complexity of one-round, information-theoretic Private Information… (More)

- Leonard J. Schulman
- IEEE Trans. Information Theory
- 1996

Let the input to a computation problem be split between two processors connected by a commu nication link and let an interactive protocol be known by which on any input the processors can solve the problem using no more than T transmissions of bits between them provided the channel is noiseless in each direction We study the following question if in fact… (More)

We provide positive and negative results concerning the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.

- Sanjoy Dasgupta, Leonard J. Schulman
- UAI
- 2000

We show that, given data from a mixture of k well-separated spherical Gaussians in !Rn, a sim ple two-round variant of EM will, with high probability, learn the centers of the Gaussians to near-optimal precision, if the dimension is high (n » log k). We relate this to previous theoreti cal and empirical work on the EM algorithm.

- Sridhar Rajagopalan, Leonard J. Schulman
- STOC
- 1994

This paper is concerned with the question of whether it is possible to sustain computation in the presence of noise; and if so, at what cost in efficiency and reliability. Shannon in his classical coding theorem showed that data can successfully and efficiently be transmitted in a noisy environment [27]. The outstanding features of the theorem are first,… (More)

- Sanjoy Dasgupta, Leonard J. Schulman
- Journal of Machine Learning Research
- 2007

We show that, given data from a mixture of k well-separated spherical Gaussians in Rd , a simple two-round variant of EM will, with high probability, learn the parameters of the Gaussians to nearoptimal precision, if the dimension is high (d lnk). We relate this to previous theoretical and empirical work on the EM algorithm.

Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function $f: X \times Y \rightarrow \{0,1\}$ and a probability distribution $D$ over $X \times Y$, we define the sampling complexity of $(f,D)$ as the minimum number of bits Alice and Bob must communicate for Alice to pick $x \in X$… (More)

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a <i>hidden subgroup problem</i>, in which a unknown subgroup <i>H</i> of a group <i>G</i> must be determined from a quantum state ψ over <i>G</i> that is uniformly supported on a left coset of <i>H</i>. These hidden subgroup problems are… (More)