1,933 research outputs found
Quantum Modeling
We present a modification of Simon's algorithm that in some cases is able to
fit experimentally obtained data to appropriately chosen trial functions with
high probability. Modulo constants pertaining to the reliability and
probability of success of the algorithm, the algorithm runs using only
O(polylog(|Y|)) queries to the quantum database and O(polylog(|X|,|Y|))
elementary quantum gates where |X| is the size of the experimental data set and
|Y| is the size of the parameter space.We discuss heuristics for good
performance, analyze the performance of the algorithm in the case of linear
regression, both one-dimensional and multidimensional, and outline the
algorithm's limitations.Comment: 16 pages, 5 figures, in Proceedings, SPIE Conference on Quantum
Computation and Quantum Information, pp. 116-127, April 21-22, 200
Online Row Sampling
Finding a small spectral approximation for a tall matrix is
a fundamental numerical primitive. For a number of reasons, one often seeks an
approximation whose rows are sampled from those of . Row sampling improves
interpretability, saves space when is sparse, and preserves row structure,
which is especially important, for example, when represents a graph.
However, correctly sampling rows from can be costly when the matrix is
large and cannot be stored and processed in memory. Hence, a number of recent
publications focus on row sampling in the streaming setting, using little more
space than what is required to store the outputted approximation [KL13,
KLM+14].
Inspired by a growing body of work on online algorithms for machine learning
and data analysis, we extend this work to a more restrictive online setting: we
read rows of one by one and immediately decide whether each row should be
kept in the spectral approximation or discarded, without ever retracting these
decisions. We present an extremely simple algorithm that approximates up to
multiplicative error and additive error using online samples, with memory overhead
proportional to the cost of storing the spectral approximation. We also present
an algorithm that uses ) memory but only requires
samples, which we show is
optimal.
Our methods are clean and intuitive, allow for lower memory usage than prior
work, and expose new theoretical properties of leverage score based matrix
approximation
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
Closing the Gap Between Short and Long XORs for Model Counting
Many recent algorithms for approximate model counting are based on a
reduction to combinatorial searches over random subsets of the space defined by
parity or XOR constraints. Long parity constraints (involving many variables)
provide strong theoretical guarantees but are computationally difficult. Short
parity constraints are easier to solve but have weaker statistical properties.
It is currently not known how long these parity constraints need to be. We
close the gap by providing matching necessary and sufficient conditions on the
required asymptotic length of the parity constraints. Further, we provide a new
family of lower bounds and the first non-trivial upper bounds on the model
count that are valid for arbitrarily short XORs. We empirically demonstrate the
effectiveness of these bounds on model counting benchmarks and in a
Satisfiability Modulo Theory (SMT) application motivated by the analysis of
contingency tables in statistics.Comment: The 30th Association for the Advancement of Artificial Intelligence
(AAAI-16) Conferenc
Weighted Birkhoff Averages and the Parameterization Method
This work provides a systematic recipe for computing accurate high order
Fourier expansions of quasiperiodic invariant circles in area preserving maps.
The recipe requires only a finite data set sampled from the quasiperiodic
circle. Our approach, being based on the parameterization method, uses a Newton
scheme to iteratively solve a conjugacy equation describing the invariant
circle. A critical step in properly formulating the conjugacy equation is to
determine the rotation number of the quasiperiodic subsystem. For this we
exploit a the weighted Birkhoff averaging method. This approach facilities
accurate computation of the rotation number given nothing but the already
mentioned orbit data.
The weighted Birkhoff averages also facilitate the computation of other
integral observables like Fourier coefficients of the parameterization of the
invariant circle. Since the parameterization method is based on a Newton
scheme, we only need to approximate a small number of Fourier coefficients with
low accuracy to find a good enough initial approximation so that Newton
converges. Moreover, the Fourier coefficients may be computed independently, so
we can sample the higher modes to guess the decay rate of the Fourier
coefficients. This allows us to choose, a-priori, an appropriate number of
modes in the truncation. We illustrate the utility of the approach for explicit
example systems including the area preserving Henon map and the standard map.
We present example computations for invariant circles with period as low as 1
and up to more than 100. We also employ a numerical continuation scheme to
compute large numbers of quasiperiodic circles in these systems. During the
continuation we monitor the Sobolev norm of the Parameterization to
automatically detect the breakdown of the family.Comment: 38 pages, 15 figure
Node counting in wireless ad-hoc networks
We study wireless ad-hoc networks consisting of small microprocessors with
limited memory, where the wireless communication between the processors can be highly unreliable. For this setting, we propose a number of algorithms to estimate the number of nodes in the network, and the number of direct neighbors of each node. The algorithms are simulated, allowing comparison of their performance
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