952 research outputs found
Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval
We prove new lower bounds for locally decodable codes and private information
retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit
alphabet, where the decoder only uses b bits of each queried position of the
codeword, needs code length m = exp(Omega(n/(2^b Sum_{i=0}^b {l choose i})))
Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries,
where the user only needs b bits from each of the two l-bit answers, unknown to
the servers, satisfies t = Omega(n/(2^b Sum_{i=0}^b {l choose i})). This
implies that several known PIR schemes are close to optimal. Our results
generalize those of Goldreich et al. who proved roughly the same bounds for
linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf, our classical
lower bounds are proved using quantum computational techniques. In particular,
we give a tight analysis of how well a 2-input function can be computed from a
quantum superposition of both inputs.Comment: 12 pages LaTeX, To appear in ICALP '0
How to Test for Diagonalizability: The Discretized PT-Invariant Square-Well Potential
Given a non-hermitean matrix M, the structure of its minimal polynomial
encodes whether M is diagonalizable or not. This note will explain how to
determine the minimal polynomial of a matrix without going through its
characteristic polynomial. The approach is applied to a quantum mechanical
particle moving in a square well under the influence of a piece-wise constant
PT-symmetric potential. Upon discretizing the configuration space, the system
is decribed by a matrix of dimension three. It turns out not to be
diagonalizable for a critical strength of the interaction, also indicated by
the transition of two real into a pair of complex energy eigenvalues. The
systems develops a three-fold degenerate eigenvalue, and two of the three
eigenfunctions disappear at this exceptional point, giving a difference between
the algebraic and geometric multiplicity of the eigenvalue equal to two.Comment: 5 page
PCA by Determinant Optimization has no Spurious Local Optima
Principal component analysis (PCA) is an indispensable tool in many learning
tasks that finds the best linear representation for data. Classically,
principal components of a dataset are interpreted as the directions that
preserve most of its "energy", an interpretation that is theoretically
underpinned by the celebrated Eckart-Young-Mirsky Theorem. There are yet other
ways of interpreting PCA that are rarely exploited in practice, largely because
it is not known how to reliably solve the corresponding non-convex optimisation
programs. In this paper, we consider one such interpretation of principal
components as the directions that preserve most of the "volume" of the dataset.
Our main contribution is a theorem that shows that the corresponding non-convex
program has no spurious local optima. We apply a number of solvers for
empirical confirmation
On the asymptotic magnitude of subsets of Euclidean space
Magnitude is a canonical invariant of finite metric spaces which has its
origins in category theory; it is analogous to cardinality of finite sets.
Here, by approximating certain compact subsets of Euclidean space with finite
subsets, the magnitudes of line segments, circles and Cantor sets are defined
and calculated. It is observed that asymptotically these satisfy the
inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex
sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in
particular, the approximation method is now known to calculate (rather than
merely define) the magnitude; also minor alterations such as references adde
Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations
Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation
The spectrum of the random environment and localization of noise
We consider random walk on a mildly random environment on finite transitive
d- regular graphs of increasing girth. After scaling and centering, the
analytic spectrum of the transition matrix converges in distribution to a
Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph
changes from a regular tree to the integers, the noise becomes localized.Comment: 18 pages, 1 figur
Some results on the Krein parameters of an association scheme
We consider association schemes with d classes and the underlying Bose-
Mesner algebra, A. Then, by taking into account the relationship between the
Hadamard and the Kronecker products of matrices and making use of some matrix
techniques over the idempotents of the unique basis of minimal orthogonal idempotents
of A , we prove some results over the Krein parameters of an association
scheme
The role of apoptosis in the development of AGM hematopoietic stem cells revealed by Bcl-2 overexpression
Apoptosis is an essential process in embryonic tissue remodeling and adult
tissue homeostasis. Within the adult hematopoietic system, it allows for
tight regulation of hematopoietic cell subsets. Previously, it was shown
that B-cell leukemia 2 (Bcl-2) overexpression in the adult increases the
viability and activity of hematopoietic cells under normal and/or
stressful conditions. However, a role for apoptosis in the embryonic
hematopoietic system has not yet been established. Since the first
hematopoietic stem cells (HSCs) are generated within the
aortagonad-mesonephros (AGM; an actively remodeling tissue) region
beginning at embryonic day 10.5, we examined this tissue for expression of
apoptosis-related genes and ongoing apoptosis. Here, we show expression of
several proapoptotic and antiapoptotic genes in the AGM. We also generated
transgenic mice overexpressing Bcl-2 under the control of the
transcriptional regulatory elements of the HSC marker stem cell antigen-1
(Sca-1), to test for the role of cell survival in the regulation of AGM
HSCs. We provide evidence for increased numbers and viability of Sca-1(+)
cells in the AGM and subdissected midgestation aortas, the site where HSCs
are localized. Most important, our in vivo transplantation data show that
Bcl-2 overexpression increases AGM and fetal liver HSC activity, strongly
suggesting that apoptosis plays a role in HSC development
Using directional curvatures to visualize folding patterns of the GTM projection manifolds
In data visualization, characterizing local geometric properties of non-linear projection manifolds provides the user with valuable additional information that can influence further steps in the data analysis. We take advantage of the smooth character of GTM projection manifold and analytically calculate its local directional curvatures. Curvature plots are useful for detecting regions where geometry is distorted, for changing the amount of regularization in non-linear projection manifolds, and for choosing regions of interest when constructing detailed lower-level visualization plots
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