Timezone: »
Spotlight
Sublinear Time LowRank Approximation of Distance Matrices
Ainesh Bakshi · David Woodruff
Let $\PP=\{ p_1, p_2, \ldots p_n \}$ and $\QQ = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\AA$ represent the $m\times n$ pairwise distance matrix with $\AA_{i,j} = d(p_i, q_j)$. Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multidimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projectioncost preserving sampling. We then show that in general, relative error approximation in sublinear time is impossible for distance matrices, even if one allows for bicriteria solutions. Additionally, we show that if $\PP = \QQ$ and $d$ is the squared Euclidean distance, which is not a metric but rather the square of a metric, then a relative error bicriteria solution can be found in sublinear time. Finally, we empirically compare our algorithm with the SVD and input sparsity time algorithms. Our algorithm is several hundred times faster than the SVD, and about $8$$20$ times faster than input sparsity methods on realworld and and synthetic datasets of size $10^8$. Accuracywise, our algorithm is only slightly worse than that of the SVD (optimal) and inputsparsity time algorithms.
Author Information
Ainesh Bakshi (Carnegie Mellon University)
David Woodruff (Carnegie Mellon University)
Related Events (a corresponding poster, oral, or spotlight)

2018 Poster: Sublinear Time LowRank Approximation of Distance Matrices »
Wed Dec 5th 03:45  05:45 PM Room Room 517 AB
More from the Same Authors

2019 Poster: Tight Dimensionality Reduction for Sketching Low Degree Polynomial Kernels »
Michela Meister · Tamas Sarlos · David Woodruff 
2019 Poster: Average Case Column Subset Selection for Entrywise $\ell_1$Norm Loss »
Zhao Song · David Woodruff · Peilin Zhong 
2019 Poster: Efficient and Thrifty Voting by Any Means Necessary »
Debmalya Mandal · Ariel Procaccia · Nisarg Shah · David Woodruff 
2019 Poster: Regularized Weighted Low Rank Approximation »
Frank Ban · David Woodruff · Richard Zhang 
2019 Oral: Efficient and Thrifty Voting by Any Means Necessary »
Debmalya Mandal · Ariel Procaccia · Nisarg Shah · David Woodruff 
2019 Poster: Total Least Squares Regression in Input Sparsity Time »
Huaian Diao · Zhao Song · David Woodruff · Xin Yang 
2019 Poster: Optimal Sketching for Kronecker Product Regression and Low Rank Approximation »
Huaian Diao · Rajesh Jayaram · Zhao Song · Wen Sun · David Woodruff 
2019 Poster: Towards a ZeroOne Law for Column Subset Selection »
Zhao Song · David Woodruff · Peilin Zhong 
2018 Poster: Robust Subspace Approximation in a Stream »
Roie Levin · Anish Prasad Sevekari · David Woodruff 
2018 Spotlight: Robust Subspace Approximation in a Stream »
Roie Levin · Anish Prasad Sevekari · David Woodruff 
2018 Poster: On Coresets for Logistic Regression »
Alexander Munteanu · Chris Schwiegelshohn · Christian Sohler · David Woodruff 
2018 Spotlight: On Coresets for Logistic Regression »
Alexander Munteanu · Chris Schwiegelshohn · Christian Sohler · David Woodruff