3,209 research outputs found
Algorithms for Approximate Subtropical Matrix Factorization
Matrix factorization methods are important tools in data mining and analysis.
They can be used for many tasks, ranging from dimensionality reduction to
visualization. In this paper we concentrate on the use of matrix factorizations
for finding patterns from the data. Rather than using the standard algebra --
and the summation of the rank-1 components to build the approximation of the
original matrix -- we use the subtropical algebra, which is an algebra over the
nonnegative real values with the summation replaced by the maximum operator.
Subtropical matrix factorizations allow "winner-takes-it-all" interpretations
of the rank-1 components, revealing different structure than the normal
(nonnegative) factorizations. We study the complexity and sparsity of the
factorizations, and present a framework for finding low-rank subtropical
factorizations. We present two specific algorithms, called Capricorn and
Cancer, that are part of our framework. They can be used with data that has
been corrupted with different types of noise, and with different error metrics,
including the sum-of-absolute differences, Frobenius norm, and Jensen--Shannon
divergence. Our experiments show that the algorithms perform well on data that
has subtropical structure, and that they can find factorizations that are both
sparse and easy to interpret.Comment: 40 pages, 9 figures. For the associated source code, see
http://people.mpi-inf.mpg.de/~pmiettin/tropical
A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs
We consider the problem of computing all-pairs shortest paths in a directed
graph with real weights assigned to vertices.
For an 0-1 matrix let be the complete weighted graph
on the rows of where the weight of an edge between two rows is equal to
their Hamming distance. Let be the weight of a minimum weight spanning
tree of
We show that the all-pairs shortest path problem for a directed graph on
vertices with nonnegative real weights and adjacency matrix can be
solved by a combinatorial randomized algorithm in time
As a corollary, we conclude that the transitive closure of a directed graph
can be computed by a combinatorial randomized algorithm in the
aforementioned time.
We also conclude that the all-pairs shortest path problem for uniform disk
graphs, with nonnegative real vertex weights, induced by point sets of bounded
density within a unit square can be solved in time
Generalizations of Bounds on the Index of Convergence to Weighted Digraphs
We study sequences of optimal walks of a growing length, in weighted
digraphs, or equivalently, sequences of entries of max-algebraic matrix powers
with growing exponents. It is known that these sequences are eventually
periodic when the digraphs are strongly connected. The transient of such
periodicity depends, in general, both on the size of digraph and on the
magnitude of the weights. In this paper, we show that some bounds on the
indices of periodicity of (unweighted) digraphs, such as the bounds of
Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply
to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure
Cyclic classes and attraction cones in max algebra
In max algebra it is well-known that the sequence A^k, with A an irreducible
square matrix, becomes periodic at sufficiently large k. This raises a number
of questions on the periodic regime of A^k and A^k x, for a given vector x.
Also, this leads to the concept of attraction cones in max algebra, by which we
mean sets of vectors whose ultimate orbit period does not exceed a given
number. This paper shows that some of these questions can be solved by matrix
squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit
period in max-min algebra. Hence the computational complexity of such problems
is of the order O(n^3 log n). The main idea is to apply an appropriate diagonal
similarity scaling A -> X^{-1}AX, called visualization scaling, and to study
the role of cyclic classes of the critical graph. For powers of a visualized
matrix in the periodic regime, we observe remarkable symmetry described by
circulants and their rectangular generalizations. We exploit this symmetry to
derive a concise system of equations for attraction cpne, and we present an
algorithm which computes the coefficients of the system.Comment: 38 page
An Overview of Transience Bounds in Max-Plus Algebra
We survey and discuss upper bounds on the length of the transient phase of
max-plus linear systems and sequences of max-plus matrix powers. In particular,
we explain how to extend a result by Nachtigall to yield a new approach for
proving such bounds and we state an asymptotic tightness result by using an
example given by Hartmann and Arguelles.Comment: 13 pages, 2 figure
Negative weights make adversaries stronger
The quantum adversary method is one of the most successful techniques for
proving lower bounds on quantum query complexity. It gives optimal lower bounds
for many problems, has application to classical complexity in formula size
lower bounds, and is versatile with equivalent formulations in terms of weight
schemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on
the principle that if an algorithm successfully computes a function then, in
particular, it is able to distinguish between inputs which map to different
values.
We present a stronger version of the adversary method which goes beyond this
principle to make explicit use of the stronger condition that the algorithm
actually computes the function. This new method, which we call ADV+-, has all
the advantages of the old: it is a lower bound on bounded-error quantum query
complexity, its square is a lower bound on formula size, and it behaves well
with respect to function composition. Moreover ADV+- is always at least as
large as the adversary method ADV, and we show an example of a monotone
function for which ADV+-(f)=Omega(ADV(f)^1.098). We also give examples showing
that ADV+- does not face limitations of ADV like the certificate complexity
barrier and the property testing barrier.Comment: 29 pages, v2: added automorphism principle, extended to non-boolean
functions, simplified examples, added matching upper bound for AD
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
On the Linear Extension Complexity of Regular n-gons
In this paper, we propose new lower and upper bounds on the linear extension
complexity of regular -gons. Our bounds are based on the equivalence between
the computation of (i) an extended formulation of size of a polytope ,
and (ii) a rank- nonnegative factorization of a slack matrix of the polytope
. The lower bound is based on an improved bound for the rectangle covering
number (also known as the boolean rank) of the slack matrix of the -gons.
The upper bound is a slight improvement of the result of Fiorini, Rothvoss and
Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp.
658-668, 2012]. The difference with their result is twofold: (i) our proof uses
a purely algebraic argument while Fiorini et al. used a geometric argument, and
(ii) we improve the base case allowing us to reduce their upper bound by one when for some integer . We conjecture that this new upper bound
is tight, which is suggested by numerical experiments for small . Moreover,
this improved upper bound allows us to close the gap with the best known lower
bound for certain regular -gons (namely, and ) hence allowing for the first time to determine their extension
complexity.Comment: 20 pages, 3 figures. New contribution: improved lower bound for the
boolean rank of the slack matrices of n-gon
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
New Dependencies of Hierarchies in Polynomial Optimization
We compare four key hierarchies for solving Constrained Polynomial
Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant
Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams
(SA) hierarchies. We prove a collection of dependencies among these hierarchies
both for general CPOPs and for optimization problems on the Boolean hypercube.
Key results include for the general case that the SONC and SOS hierarchy are
polynomially incomparable, while SDSOS is contained in SONC. A direct
consequence is the non-existence of a Putinar-like Positivstellensatz for
SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like
versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent.
Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that
provides a O(n) degree bound.Comment: 26 pages, 4 figure
- âŠ