Evidence accumulation in a Laplace domain decision space

Evidence accumulation models of simple decision-making have long assumed that the brain estimates a scalar decision variable corresponding to the log-likelihood ratio of the two alternatives. Typical neural implementations of this algorithmic cognitive model assume that large numbers of neurons are each noisy exemplars of the scalar decision variable. Here we propose a neural implementation of the diffusion model in which many neurons construct and maintain the Laplace transform of the distance to each of the decision bounds. As in classic findings from brain regions including LIP, the firing rate of neurons coding for the Laplace transform of net accumulated evidence grows to a bound during random dot motion tasks. However, rather than noisy exemplars of a single mean value, this approach makes the novel prediction that firing rates grow to the bound exponentially, across neurons there should be a distribution of different rates. A second set of neurons records an approximate inversion of the Laplace transform, these neurons directly estimate net accumulated evidence. In analogy to time cells and place cells observed in the hippocampus and other brain regions, the neurons in this second set have receptive fields along a "decision axis." This finding is consistent with recent findings from rodent recordings. This theoretical approach places simple evidence accumulation models in the same mathematical language as recent proposals for representing time and space in cognitive models for memory.Comment: Revised for CB

Logic analysis of complex systems by characterizing failure phenomena to achieve diagnosis and fault-isolation

A recent result shows that, for a certain class of systems, the interdependency among the elements of such a system together with the elements constitutes a mathematical structure a partially ordered set. It is called a loop free logic model of the system. On the basis of an intrinsic property of the mathematical structure, a characterization of system component failure in terms of maximal subsets of bad test signals of the system was obtained. Also, as a consequence, information concerning the total number of failure components in the system was deduced. Detailed examples are given to show how to restructure real systems containing loops into loop free models for which the result is applicable

Sparse Estimation with the Swept Approximated Message-Passing Algorithm

Approximate Message Passing (AMP) has been shown to be a superior method for inference problems, such as the recovery of signals from sets of noisy, lower-dimensionality measurements, both in terms of reconstruction accuracy and in computational efficiency. However, AMP suffers from serious convergence issues in contexts that do not exactly match its assumptions. We propose a new approach to stabilizing AMP in these contexts by applying AMP updates to individual coefficients rather than in parallel. Our results show that this change to the AMP iteration can provide theoretically expected, but hitherto unobtainable, performance for problems on which the standard AMP iteration diverges. Additionally, we find that the computational costs of this swept coefficient update scheme is not unduly burdensome, allowing it to be applied efficiently to signals of large dimensionality.Comment: 11 pages, 3 figures, implementation available at https://github.com/eric-tramel/SwAMP-Dem

Hard Matching for Boosted Tops at Two Loops

Cross sections for top quarks provide very interesting physics opportunities, being both sensitive to new physics and also perturbatively tractable due to the large top quark mass. Rigorous factorization theorems for top cross sections can be derived in several kinematic scenarios, including the boosted regime in the peak region that we consider here. In the context of the corresponding factorization theorem for $e^+e^-$ collisions we extract the last missing ingredient that is needed to evaluate the cross section differential in the jet-mass at two-loop order, namely the matching coefficient at the scale $\mu\simeq m_t$. Our extraction also yields the final ingredients needed to carry out logarithmic resummation at next-to-next-to-leading logarithmic order (or N$^3$LL if we ignore the missing 4-loop cusp anomalous dimension). This coefficient exhibits an amplitude level rapidity logarithm starting at $\mathcal{O}(\alpha_s^2)$ due to virtual top quark loops, which we treat using rapidity renormalization group (RG) evolution. Interestingly, this rapidity RG evolution appears in the matching coefficient between two effective theories around the heavy quark mass scale $\mu\simeq m_t$.Comment: 35 pages, 3 figures, v2: added extraction of 3-loop anon. dimension, journal versio

Infrared Renormalization Group Flow for Heavy Quark Masses

A short-distance heavy quark mass depends on two parameters, the renormalization scale mu controlling the absorption of ultraviolet fluctuations into the mass, and a scale R controlling the absorption of infrared fluctuations. 1/R can be thought of as the radius for perturbative corrections that build up the mass beyond its point-like definition in the pole scheme. Treating R as a variable gives a renormalization group equation. We argue that the sign of this anomalous dimension is universal: increasing R to add IR modes decreases m(R). The flow improves the stability of conversions between mass schemes, allowing us to avoid large logs and the renormalon. The flow in R can be used to study IR renormalons without using bubble chains, and we use it to determine the coefficient of the LambdaQCD renormalon ambiguity of the pole mass with a convergent sum-rule.Comment: 4 pages, 2 figures, Added explicit result for the top MSbar mass with uncertaintie

Factorization Approach for Top Mass Reconstruction at High Energies

Using effective theories for jets and heavy quarks it is possible to prove that the double differential top-antitop invariant mass distribution for the process $e^+e^-\to t\bar t$ in the resonance region for c.m. energies $Q$ much larger than the top mass can factorized into perturbatively computable hard coefficients and jet functions and a non-perturbative soft function. For invariant mass prescriptions based on hemispheres defined with respect to the thrust axis the soft function can be extracted from massless jet event shape distributions. This approach allows in principle for top mass determinations without uncertainties from hadronization using the reconstruction method and to quantify the top mass scheme dependence of the measured top quark mass value.Comment: Talk given at 2007 International Linear Collider Workshop (LCWS07 and ILC07), Hamburg, Germany, 30 May - 3 Jun 2007, 7 pages, 4 figures, title modifie