20,108 research outputs found
Analysis of a Classical Matrix Preconditioning Algorithm
We study a classical iterative algorithm for balancing matrices in the
norm via a scaling transformation. This algorithm, which goes back
to Osborne and Parlett \& Reinsch in the 1960s, is implemented as a standard
preconditioner in many numerical linear algebra packages. Surprisingly, despite
its widespread use over several decades, no bounds were known on its rate of
convergence. In this paper we prove that, for any irreducible (real
or complex) input matrix~, a natural variant of the algorithm converges in
elementary balancing operations, where
measures the initial imbalance of~ and is the target imbalance
of the output matrix. (The imbalance of~ is , where
are the maximum entries in magnitude in the
th row and column respectively.) This bound is tight up to the
factor. A balancing operation scales the th row and column so that their
maximum entries are equal, and requires arithmetic operations on
average, where is the number of non-zero elements in~. Thus the running
time of the iterative algorithm is . This is the first time
bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. We
also prove a conjecture of Chen that characterizes those matrices for which the
limit of the balancing process is independent of the order in which balancing
operations are performed.Comment: The previous version (1) (see also STOC'15) handled UB ("unique
balance") input matrices. In this version (2) we extend the work to handle
all input matrice
Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
An important theorem of Banaszczyk (Random Structures & Algorithms `98)
states that for any sequence of vectors of norm at most and any
convex body of Gaussian measure in , there exists a
signed combination of these vectors which lands inside . A major open
problem is to devise a constructive version of Banaszczyk's vector balancing
theorem, i.e. to find an efficient algorithm which constructs the signed
combination.
We make progress towards this goal along several fronts. As our first
contribution, we show an equivalence between Banaszczyk's theorem and the
existence of -subgaussian distributions over signed combinations. For the
case of symmetric convex bodies, our equivalence implies the existence of a
universal signing algorithm (i.e. independent of the body), which simply
samples from the subgaussian sign distribution and checks to see if the
associated combination lands inside the body. For asymmetric convex bodies, we
provide a novel recentering procedure, which allows us to reduce to the case
where the body is symmetric.
As our second main contribution, we show that the above framework can be
efficiently implemented when the vectors have length ,
recovering Banaszczyk's results under this stronger assumption. More precisely,
we use random walk techniques to produce the required -subgaussian
signing distributions when the vectors have length , and
use a stochastic gradient ascent method to implement the recentering procedure
for asymmetric bodies
Extremal Problems in Minkowski Space related to Minimal Networks
We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan
[FLM]: Is there an upper bound polynomial in for the largest cardinality of
a set S of unit vectors in an n-dimensional Minkowski space (or Banach space)
such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n
and that equality holds iff the space is linearly isometric to \ell^n_\infty,
the space with an n-cube as unit ball. We also remark on similar questions
raised in [FLM] that arose out of the study of singularities in
length-minimizing networks in Minkowski spaces.Comment: 6 pages. 11-year old paper. Implicit question in the last sentence
has been answered in Discrete & Computational Geometry 21 (1999) 437-44
Balancing sums of random vectors
We study a higher-dimensional 'balls-into-bins' problem. An infinite sequence
of i.i.d. random vectors is revealed to us one vector at a time, and we are
required to partition these vectors into a fixed number of bins in such a way
as to keep the sums of the vectors in the different bins close together; how
close can we keep these sums almost surely? This question, our primary focus in
this paper, is closely related to the classical problem of partitioning a
sequence of vectors into balanced subsequences, in addition to having
applications to some problems in computer science.Comment: 17 pages, Discrete Analysi
Symbol-Level Precoding Design for Max-Min SINR in Multiuser MISO Broadcast Channels
In this paper, we address the symbol level precoding (SLP) design problem
under max-min SINR criterion in the downlink of multiuser multiple-input
single-output (MISO) channels. First, we show that the distance preserving
constructive interference regions (DPCIR) are always polyhedral angles (shifted
pointed cones) for any given constellation point with unbounded decision
region. Then we prove that any signal in a given unbounded DPCIR has a norm
larger than the norm of the corresponding vertex if and only if the convex hull
of the constellation contains the origin. Using these properties, we show that
the power of the noiseless received signal lying on an unbounded DPCIR is an
strictly increasing function of two parameters. This allows us to reformulate
the originally non-convex SLP max-min SINR as a convex optimization problem. We
discuss the loss due to our proposed convex reformulation and provide some
simulation results.Comment: Submitted to SPAWC 2018, 7 pages, 2 figure
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