71,404 research outputs found
Counting to Ten with Two Fingers: Compressed Counting with Spiking Neurons
We consider the task of measuring time with probabilistic threshold gates implemented by bio-inspired spiking neurons. In the model of spiking neural networks, network evolves in discrete rounds, where in each round, neurons fire in pulses in response to a sufficiently high membrane potential. This potential is induced by spikes from neighboring neurons that fired in the previous round, which can have either an excitatory or inhibitory effect.
Discovering the underlying mechanisms by which the brain perceives the duration of time is one of the largest open enigma in computational neuro-science. To gain a better algorithmic understanding onto these processes, we introduce the neural timer problem. In this problem, one is given a time parameter t, an input neuron x, and an output neuron y. It is then required to design a minimum sized neural network (measured by the number of auxiliary neurons) in which every spike from x in a given round i, makes the output y fire for the subsequent t consecutive rounds.
We first consider a deterministic implementation of a neural timer and show that Theta(log t) (deterministic) threshold gates are both sufficient and necessary. This raised the question of whether randomness can be leveraged to reduce the number of neurons. We answer this question in the affirmative by considering neural timers with spiking neurons where the neuron y is required to fire for t consecutive rounds with probability at least 1-delta, and should stop firing after at most 2t rounds with probability 1-delta for some input parameter delta in (0,1). Our key result is a construction of a neural timer with O(log log 1/delta) spiking neurons. Interestingly, this construction uses only one spiking neuron, while the remaining neurons can be deterministic threshold gates. We complement this construction with a matching lower bound of Omega(min{log log 1/delta, log t}) neurons. This provides the first separation between deterministic and randomized constructions in the setting of spiking neural networks.
Finally, we demonstrate the usefulness of compressed counting networks for synchronizing neural networks. In the spirit of distributed synchronizers [Awerbuch-Peleg, FOCS\u2790], we provide a general transformation (or simulation) that can take any synchronized network solution and simulate it in an asynchronous setting (where edges have arbitrary response latencies) while incurring a small overhead w.r.t the number of neurons and computation time
Sparse Recovery with Very Sparse Compressed Counting
Compressed sensing (sparse signal recovery) often encounters nonnegative data
(e.g., images). Recently we developed the methodology of using (dense)
Compressed Counting for recovering nonnegative K-sparse signals. In this paper,
we adopt very sparse Compressed Counting for nonnegative signal recovery. Our
design matrix is sampled from a maximally-skewed p-stable distribution (0<p<1),
and we sparsify the design matrix so that on average (1-g)-fraction of the
entries become zero. The idea is related to very sparse stable random
projections (Li et al 2006 and Li 2007), the prior work for estimating summary
statistics of the data.
In our theoretical analysis, we show that, when p->0, it suffices to use M=
K/(1-exp(-gK) log N measurements, so that all coordinates can be recovered in
one scan of the coordinates. If g = 1 (i.e., dense design), then M = K log N.
If g= 1/K or 2/K (i.e., very sparse design), then M = 1.58K log N or M = 1.16K
log N. This means the design matrix can be indeed very sparse at only a minor
inflation of the sample complexity.
Interestingly, as p->1, the required number of measurements is essentially M
= 2.7K log N, provided g= 1/K. It turns out that this result is a general
worst-case bound
Estimating Entropy of Data Streams Using Compressed Counting
The Shannon entropy is a widely used summary statistic, for example, network
traffic measurement, anomaly detection, neural computations, spike trains, etc.
This study focuses on estimating Shannon entropy of data streams. It is known
that Shannon entropy can be approximated by Reenyi entropy or Tsallis entropy,
which are both functions of the p-th frequency moments and approach Shannon
entropy as p->1.
Compressed Counting (CC) is a new method for approximating the p-th frequency
moments of data streams. Our contributions include:
1) We prove that Renyi entropy is (much) better than Tsallis entropy for
approximating Shannon entropy.
2) We propose the optimal quantile estimator for CC, which considerably
improves the previous estimators.
3) Our experiments demonstrate that CC is indeed highly effective
approximating the moments and entropies. We also demonstrate the crucial
importance of utilizing the variance-bias trade-off
Experimentally exploring compressed sensing quantum tomography
In the light of the progress in quantum technologies, the task of verifying
the correct functioning of processes and obtaining accurate tomographic
information about quantum states becomes increasingly important. Compressed
sensing, a machinery derived from the theory of signal processing, has emerged
as a feasible tool to perform robust and significantly more resource-economical
quantum state tomography for intermediate-sized quantum systems. In this work,
we provide a comprehensive analysis of compressed sensing tomography in the
regime in which tomographically complete data is available with reliable
statistics from experimental observations of a multi-mode photonic
architecture. Due to the fact that the data is known with high statistical
significance, we are in a position to systematically explore the quality of
reconstruction depending on the number of employed measurement settings,
randomly selected from the complete set of data, and on different model
assumptions. We present and test a complete prescription to perform efficient
compressed sensing and are able to reliably use notions of model selection and
cross-validation to account for experimental imperfections and finite counting
statistics. Thus, we establish compressed sensing as an effective tool for
quantum state tomography, specifically suited for photonic systems.Comment: 12 pages, 5 figure
The Hierarchical - Trajectory by Perturbation Theory in a Running Coupling and its Logarithm
We compute the hierarchical -trajectory in terms of perturbation
theory in a running coupling. In the three dimensional case we resolve a
singularity due to resonance of power counting factors in terms of logarithms
of the running coupling. Numerical data is presented and the limits of validity
explored. We also compute moving eigenvalues and eigenvectors on the trajectory
as well as their fusion rules.Comment: 24 pages, 9 pictures included, uuencoded compressed postscript fil
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