13,049 research outputs found
Low-Rank Matrix Approximation with Weights or Missing Data is NP-hard
Weighted low-rank approximation (WLRA), a dimensionality reduction technique
for data analysis, has been successfully used in several applications, such as
in collaborative filtering to design recommender systems or in computer vision
to recover structure from motion. In this paper, we study the computational
complexity of WLRA and prove that it is NP-hard to find an approximate
solution, even when a rank-one approximation is sought. Our proofs are based on
a reduction from the maximum-edge biclique problem, and apply to strictly
positive weights as well as binary weights (the latter corresponding to
low-rank matrix approximation with missing data).Comment: Proof of Lemma 4 (Lemma 3 in v1) has been corrected. Some remarks and
comments have been added. Accepted in SIAM Journal on Matrix Analysis and
Application
Quantified CTL: Expressiveness and Complexity
While it was defined long ago, the extension of CTL with quantification over
atomic propositions has never been studied extensively. Considering two
different semantics (depending whether propositional quantification refers to
the Kripke structure or to its unwinding tree), we study its expressiveness
(showing in particular that QCTL coincides with Monadic Second-Order Logic for
both semantics) and characterise the complexity of its model-checking and
satisfiability problems, depending on the number of nested propositional
quantifiers (showing that the structure semantics populates the polynomial
hierarchy while the tree semantics populates the exponential hierarchy)
Asymptotic of geometrical navigation on a random set of points of the plane
A navigation on a set of points is a rule for choosing which point to
move to from the present point in order to progress toward a specified target.
We study some navigations in the plane where is a non uniform Poisson point
process (in a finite domain) with intensity going to . We show the
convergence of the traveller path lengths, the number of stages done, and the
geometry of the traveller trajectories, uniformly for all starting points and
targets, for several navigations of geometric nature. Other costs are also
considered. This leads to asymptotic results on the stretch factors of random
Yao-graphs and random -graphs
Almost harmonic spinors
We show that any closed spin manifold not diffeomorphic to the two-sphere
admits a sequence of volume-one-Riemannian metrics for which the smallest
non-zero Dirac eigenvalue tends to zero. As an application, we compare the
Dirac spectrum with the conformal volume.Comment: minor modifications of the published versio
The combinatorics of the colliding bullets problem
The finite colliding bullets problem is the following simple problem:
consider a gun, whose barrel remains in a fixed direction; let be an i.i.d.\ family of random variables with uniform distribution on
; shoot bullets one after another at times , where the
th bullet has speed . When two bullets collide, they both annihilate.
We give the distribution of the number of surviving bullets, and in some
generalisation of this model. While the distribution is relatively simple (and
we found a number of bold claims online), our proof is surprisingly intricate
and mixes combinatorial and geometric arguments; we argue that any rigorous
argument must very likely be rather elaborate.Comment: 29 page
Nonnegative factorization and the maximum edge biclique problem
Nonnegative matrix factorization (NMF) is a data analysis technique based on the approximation of a nonnegative matrix with a product of two nonnegative factors, which allows compression and interpretation of nonnegative data. In this paper, we study the case of rank-one factorization and show that when the matrix to be factored is not required to be nonnegative, the corresponding problem (R1NF) becomes NP-hard. This sheds new light on the complexity of NMF since any algorithm for fixed-rank NMF must be able to solve at least implicitly such rank-one subproblems. Our proof relies on a reduction of the maximum edge biclique problem to R1NF. We also link stationary points of R1NF to feasible solutions of the biclique problem, which allows us to design a new type of biclique finding algorithm based on the application of a block-coordinate descent scheme to R1NF. We show that this algorithm, whose algorithmic complexity per iteration is proportional to the number of edges in the graph, is guaranteed to converge to a biclique and that it performs competitively with existing methods on random graphs and text mining datasets.nonnegative matrix factorization, rank-one factorization, maximum edge biclique problem, algorithmic complexity, biclique finding algorithm
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