Practical Procedures for Dimension Reduction in l_1

Practical Procedures for Dimension Reduction in l_1

Cached

Download Links

[www-stat.stanford.edu]

by Ping Li , Trevor J. Hastie , Kenneth W. Church

BibTeX

@MISC{Li_practicalprocedures,
    author = {Ping Li and Trevor J. Hastie and Kenneth W. Church},
    title = {Practical Procedures for Dimension Reduction in l_1},
    year = {}
}

Bookmark

OpenURL

Abstract

We show that an analog of the Johnson-Lindestrauss (JL) lemma for dimension reduction in l 1 can be established using linear projections and nonlinear estimators. Previous studies have proved that no JL lemma exists for l 1 using linear estimators. We develop two nonlinear estimators including a strictly unbiased estimator and an improved estimator based on the maximum likelihood. While the maximum likelihood estimator (MLE) does not have a closed-form density function, we propose highly accurate closed-form approximations.

Citations

999	Regression shrinkage and selection via the lasso - Tibshirani - 2006
843	Information theory and an extension of the maximum likelihood principle - Akaike - 1973
825	A: Categorical data analysis - Agresti - 1990
291	Extensions of Lipschitz mapping into Hilbert space - Johnson, Lindenstrauss - 1984
254	On the resemblance and containment of documents - Broder - 1997
231	Concept decompositions for large sparse text data using clustering - Dhillon, Modha - 2001
207	Using the nyström method to speed up kernel machines - Williams, Seeger
206	Exploratory projection pursuit - Friedman - 1987
205	Stable distributions, pseudorandom generators, embeddings, and data stream computation - Indyk
175	Fingerprinting by random polynomials - Rabin - 1981
151	Min-wise Independent Permutations - Broder, Charikar, et al. - 1998
138	Learning Mixtures of Gaussians - Dasgupta - 1999
135	Similarity estimation techniques from rounding algorithms - Charikar - 2002
133	Efficient svm training using lowrank kernel representation - Fine, Scheinberg
113	Power laws, Pareto distributions and Zipf’s law - Newman - 2005
111	Support vector machines for histogram-based image classification - Chapelle, Haffner, et al. - 1999
101	Algorithmic applications of low-distortion geometric embeddings - Indyk - 2001
99	Random projection in dimensionality reduction: applications to image and text data - Bingham, Mannila - 2001
93	Dimensionality reduction by random mapping: Fast similarity computation for clustering - Kaski - 1998
80	Database-friendly random projections: Johnson-Lindenstrauss with binary coins - Achlioptas - 2003
74	The Johnson-Lindenstrauss Lemma and the sphericity of some graphs - Frankl, Maehara - 1987
58	Embedding the diamond graph in Lp and dimension reduction in L1 - Lee, Naor - 1995
56	On the nyström method for approximating a gram matrix for improved kernel-based learning - Drineas, Mahoney
50	Simple Consistent Estimators of Stable Distribution - McCulloch - 1986
46	Testing Statistical Hypothesis - Lehmann - 1986
45	The Random Projection Method - Vempala - 2005
42	Sampling techniques for kernel methods - Achlioptas, McSherry, et al. - 2002
38	Survival models for heterogenous populations derived from stable distributions. Biometrika 73 - Hougaard - 1986
38	Building support vector machines with reduced classifier complexity - Keerthi, Chapelle, et al.
37	On the impossibility of dimension reduction in l1 - Brinkman, Charikar - 2005
36	On spectral learning of mixtures of distributions - Achlioptas, McSherry
34	Parameter estimates for symmetric stable distributions - Fama, Roll - 1971
31	A spectral algorithm for learning mixtures of distributions - Vempala, Wang - 2002
30	Min-wise independent permutations (extended abstract - Broder, Charikar, et al. - 1998
30	Some properties of symmetric stable distributions - Fama, Roll - 1968
29	Two new properties of mathematical likelihood - Fisher - 1934
22	A scalable approach to balanced, high-dimensional clustering of market-baskets - Strehl, Ghosh - 2000
16	Approximate confidence intervals - Bartlett - 1953
16	A derandomization using minwise independent permutations - Broder, Charikar, et al. - 1998
12	Aleksandrs Slivkins. Metric embeddings with relaxed guarantees - Abraham, Bartal, et al. - 2005
11	The Inverse Gaussian Distribution. A case study in exponential families - Seshadri - 1994
10	Higher moments of a maximum-likelihood estimate - Shenton, Bowman - 1963
8	Inferences for the cauchy distribution based on maximum likelihood estimators - Haas, Bain, et al. - 1970
8	Using sketches to estimate associations - Li, Church - 2005
8	On the Distribution of SINR for the MMSE MIMO Receiver and Performance Analysis - Li, Paul, et al. - 2006
6	Evaluation of the maximum-likelihood estimator where the likelihood equation has multiple roots - Barnett - 1966
5	Hilbertian metrics on probability measures and their application in SVMs - Hein, Lal, et al. - 2004
5	Likelihood inference about location and scale parameters - Hinkley - 1978
5	Statistical properties of Inverse Gaussian distributions - Tweedie - 1957
3	Skewness and kurtosis for maximum likelihood estimator in one-parameter exponential family models - Cysneiros, Santos, et al. - 1999