180,059 research outputs found
Pattern Avoidance for Random Permutations
Using techniques from Poisson approximation, we prove explicit error bounds
on the number of permutations that avoid any pattern. Most generally, we bound
the total variation distance between the joint distribution of pattern
occurrences and a corresponding joint distribution of independent Bernoulli
random variables, which as a corollary yields a Poisson approximation for the
distribution of the number of occurrences of any pattern. We also investigate
occurrences of consecutive patterns in random Mallows permutations, of which
uniform random permutations are a special case. These bounds allow us to
estimate the probability that a pattern occurs any number of times and, in
particular, the probability that a random permutation avoids a given pattern.Comment: 24 pages, 2 Figures, 4 Table
Learning Fair Naive Bayes Classifiers by Discovering and Eliminating Discrimination Patterns
As machine learning is increasingly used to make real-world decisions, recent
research efforts aim to define and ensure fairness in algorithmic decision
making. Existing methods often assume a fixed set of observable features to
define individuals, but lack a discussion of certain features not being
observed at test time. In this paper, we study fairness of naive Bayes
classifiers, which allow partial observations. In particular, we introduce the
notion of a discrimination pattern, which refers to an individual receiving
different classifications depending on whether some sensitive attributes were
observed. Then a model is considered fair if it has no such pattern. We propose
an algorithm to discover and mine for discrimination patterns in a naive Bayes
classifier, and show how to learn maximum likelihood parameters subject to
these fairness constraints. Our approach iteratively discovers and eliminates
discrimination patterns until a fair model is learned. An empirical evaluation
on three real-world datasets demonstrates that we can remove exponentially many
discrimination patterns by only adding a small fraction of them as constraints
Lower Bounds for Oblivious Near-Neighbor Search
We prove an lower bound on the dynamic
cell-probe complexity of statistically
approximate-near-neighbor search () over the -dimensional
Hamming cube. For the natural setting of , our result
implies an lower bound, which is a quadratic
improvement over the highest (non-oblivious) cell-probe lower bound for
. This is the first super-logarithmic
lower bound for against general (non black-box) data structures.
We also show that any oblivious data structure for
decomposable search problems (like ) can be obliviously dynamized
with overhead in update and query time, strengthening a classic
result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page
Quantum key distribution with an efficient countermeasure against correlated intensity fluctuations in optical pulses
Quantum key distribution (QKD) allows two distant parties to share secret
keys with the proven security even in the presence of an eavesdropper with
unbounded computational power. Recently, GHz-clock decoy QKD systems have been
realized by employing ultrafast optical communication devices. However,
security loopholes of high-speed systems have not been fully explored yet. Here
we point out a security loophole at the transmitter of the GHz-clock QKD, which
is a common problem in high-speed QKD systems using practical band-width
limited devices. We experimentally observe the inter-pulse intensity
correlation and modulation-pattern dependent intensity deviation in a practical
high-speed QKD system. Such correlation violates the assumption of most
security theories. We also provide its countermeasure which does not require
significant changes of hardware and can generate keys secure over 100 km fiber
transmission. Our countermeasure is simple, effective and applicable to wide
range of high-speed QKD systems, and thus paves the way to realize ultrafast
and security-certified commercial QKD systems
Error counting in a quantum error-correcting code and the ground-state energy of a spin glass
Upper and lower bounds are given for the number of equivalence classes of
error patterns in the toric code for quantum memory. The results are used to
derive a lower bound on the ground-state energy of the +/-J Ising spin glass
model on the square lattice with symmetric and asymmetric bond distributions.
This is a highly non-trivial example in which insights from quantum information
lead directly to an explicit result on a physical quantity in the statistical
mechanics of disordered systems.Comment: 15 pages, 7 figures, JPSJ style, latex style file include
Characterizing Pixel and Point Patterns with a Hyperuniformity Disorder Length
We introduce the concept of a hyperuniformity disorder length that controls
the variance of volume fraction fluctuations for randomly placed windows of
fixed size. In particular, fluctuations are determined by the average number of
particles within a distance from the boundary of the window. We first
compute special expectations and bounds in dimensions, and then illustrate
the range of behavior of versus window size by analyzing three
different types of simulated two-dimensional pixel pattern - where particle
positions are stored as a binary digital image in which pixels have value
zero/one if empty/contain a particle. The first are random binomial patterns,
where pixels are randomly flipped from zero to one with probability equal to
area fraction. These have long-ranged density fluctuations, and simulations
confirm the exact result . Next we consider vacancy patterns, where a
fraction of particles on a lattice are randomly removed. These also display
long-range density fluctuations, but with for small . For a
hyperuniform system with no long-range density fluctuations, we consider
Einstein patterns where each particle is independently displaced from a lattice
site by a Gaussian-distributed amount. For these, at large , approaches
a constant equal to about half the root-mean-square displacement in each
dimension. Then we turn to grayscale pixel patterns that represent simulated
arrangements of polydisperse particles, where the volume of a particle is
encoded in the value of its central pixel. And we discuss the continuum limit
of point patterns, where pixel size vanishes. In general, we thus propose to
quantify particle configurations not just by the scaling of the density
fluctuation spectrum but rather by the real-space spectrum of versus
. We call this approach Hyperuniformity Disorder Length Spectroscopy
Formulas vs. Circuits for Small Distance Connectivity
We give the first super-polynomial separation in the power of bounded-depth
boolean formulas vs. circuits. Specifically, we consider the problem Distance
Connectivity, which asks whether two specified nodes in a graph of size
are connected by a path of length at most . This problem is solvable
(by the recursive doubling technique) on {\bf circuits} of depth
and size . In contrast, we show that solving this problem on {\bf
formulas} of depth requires size for all . As corollaries:
(i) It follows that polynomial-size circuits for Distance Connectivity
require depth for all . This matches the
upper bound from recursive doubling and improves a previous lower bound of Beame, Pitassi and Impagliazzo [BIP98].
(ii) We get a tight lower bound of on the size required to
simulate size- depth- circuits by depth- formulas for all and . No lower bound better than
was previously known for any .
Our proof technique is centered on a new notion of pathset complexity, which
roughly speaking measures the minimum cost of constructing a set of (partial)
paths in a universe of size via the operations of union and relational
join, subject to certain density constraints. Half of our proof shows that
bounded-depth formulas solving Distance Connectivity imply upper bounds
on pathset complexity. The other half is a combinatorial lower bound on pathset
complexity
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