3,198 research outputs found
Toward a unified theory of sparse dimensionality reduction in Euclidean space
Let be a sparse Johnson-Lindenstrauss
transform [KN14] with non-zeroes per column. For a subset of the unit
sphere, given, we study settings for required to
ensure i.e. so that preserves the norm of every
simultaneously and multiplicatively up to . We
introduce a new complexity parameter, which depends on the geometry of , and
show that it suffices to choose and such that this parameter is small.
Our result is a sparse analog of Gordon's theorem, which was concerned with a
dense having i.i.d. Gaussian entries. We qualitatively unify several
results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and
Fourier-based restricted isometries. Our work also implies new results in using
the sparse Johnson-Lindenstrauss transform in numerical linear algebra,
classical and model-based compressed sensing, manifold learning, and
constrained least squares problems such as the Lasso
Input Sparsity and Hardness for Robust Subspace Approximation
In the subspace approximation problem, we seek a k-dimensional subspace F of
R^d that minimizes the sum of p-th powers of Euclidean distances to a given set
of n points a_1, ..., a_n in R^d, for p >= 1. More generally than minimizing
sum_i dist(a_i,F)^p,we may wish to minimize sum_i M(dist(a_i,F)) for some loss
function M(), for example, M-Estimators, which include the Huber and Tukey loss
functions. Such subspaces provide alternatives to the singular value
decomposition (SVD), which is the p=2 case, finding such an F that minimizes
the sum of squares of distances. For p in [1,2), and for typical M-Estimators,
the minimizing gives a solution that is more robust to outliers than that
provided by the SVD. We give several algorithmic and hardness results for these
robust subspace approximation problems.
We think of the n points as forming an n x d matrix A, and letting nnz(A)
denote the number of non-zero entries of A. Our results hold for p in [1,2). We
use poly(n) to denote n^{O(1)} as n -> infty. We obtain: (1) For minimizing
sum_i dist(a_i,F)^p, we give an algorithm running in O(nnz(A) +
(n+d)poly(k/eps) + exp(poly(k/eps))), (2) we show that the problem of
minimizing sum_i dist(a_i, F)^p is NP-hard, even to output a
(1+1/poly(d))-approximation, answering a question of Kannan and Vempala, and
complementing prior results which held for p >2, (3) For loss functions for a
wide class of M-Estimators, we give a problem-size reduction: for a parameter
K=(log n)^{O(log k)}, our reduction takes O(nnz(A) log n + (n+d) poly(K/eps))
time to reduce the problem to a constrained version involving matrices whose
dimensions are poly(K eps^{-1} log n). We also give bicriteria solutions, (4)
Our techniques lead to the first O(nnz(A) + poly(d/eps)) time algorithms for
(1+eps)-approximate regression for a wide class of convex M-Estimators.Comment: paper appeared in FOCS, 201
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