Learning to detect a tone in unpredictable noise

Eight normal-hearing listeners practiced a tone-detection task in which a 1-kHz target was masked by a spectrally unpredictable multitone complex. Consistent learning was observed, with mean masking decreasing by 6.4 dB over five sessions (4500 trials). Reverse-correlation was used to estimate how listeners weighted each spectral region. Weight-vectors approximated the ideal more closely after practice, indicating that listeners were learning to attend selectively to the task relevant information. Once changes in weights were accounted for, no changes in internal noise (psychometric slope) were observed. It is concluded that this task elicits robust learning, which can be understood primarily as improved selective attention

The Role of Response Bias in Perceptual Learning

Sensory judgments improve with practice. Such perceptual learning is often thought to reflect an increase in perceptual sensitivity. However, it may also represent a decrease in response bias, with unpracticed observers acting in part on a priori hunches rather than sensory evidence. To examine whether this is the case, 55 observers practiced making a basic auditory judgment (yes/no amplitude-modulation detection or forced-choice frequency/amplitude discrimination) over multiple days. With all tasks, bias was present initially, but decreased with practice. Notably, this was the case even on supposedly “bias-free,” 2-alternative forced-choice, tasks. In those tasks, observers did not favor the same response throughout (stationary bias), but did favor whichever response had been correct on previous trials (nonstationary bias). Means of correcting for bias are described. When applied, these showed that at least 13% of perceptual learning on a forced-choice task was due to reduction in bias. In other situations, changes in bias were shown to obscure the true extent of learning, with changes in estimated sensitivity increasing once bias was corrected for. The possible causes of bias and the implications for our understanding of perceptual learning are discussed

Reduction of internal noise in auditory perceptual learning

This paper examines what mechanisms underlie auditory perceptual learning. Fifteen normal hearing adults performed two-alternative, forced choice, pure tone frequency discrimination for four sessions. External variability was introduced by adding a zero-mean Gaussian random variable to the frequency of each tone. Measures of internal noise, encoding efficiency, bias, and inattentiveness were derived using four methods (model fit, classification boundary, psychometric function, and double-pass consistency). The four methods gave convergent estimates of internal noise, which was found to decrease from 4.52 to 2.93 Hz with practice. No group-mean changes in encoding efficiency, bias, or inattentiveness were observed. It is concluded that learned improvements in frequency discrimination primarily reflect a reduction in internal noise. Data from highly experienced listeners and neural networks performing the same task are also reported. These results also indicated that auditory learning represents internal noise reduction, potentially through the re-weighting of frequency-specific channels

Stochastic stability at the boundary of expanding maps

We consider endomorphisms of a compact manifold which are expanding except for a finite number of points and prove the existence and uniqueness of a physical measure and its stochastical stability. We also characterize the zero-noise limit measures for a model of the intermittent map and obtain stochastic stability for some values of the parameter. The physical measures are obtained as zero-noise limits which are shown to satisfy Pesin?s Entropy Formula

Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.Comment: To appear in Journal of Statistical Physic

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3

We consider an Euclidean supersymmetric field theory in $Z^3$ given by a supersymmetric $\Phi^4$ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green's function of a (stable) L\'evy random walk in $Z^3$. The Green's function depends on the L\'evy-Khintchine parameter $\alpha={3+\epsilon\over 2}$ with $0<\alpha<2$. For $\alpha ={3\over 2}$ the $\Phi^{4}$ interaction is marginal. We prove for $\alpha-{3\over 2}={\epsilon\over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green's function of a (weakly) self-avoiding L\'evy walk in $Z^3$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The control of the renormalization group trajectory is a preparation for the study of the asymptotics of this Green's function. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding L\'evy walk in $Z^3$.Comment: 82 pages, Tex with macros supplied. Revision includes 1. redefinition of norms involving fermions to ensure uniqueness. 2. change in the definition of lattice blocks and lattice polymer activities. 3. Some proofs have been reworked. 4. New lemmas 5.4A, 5.14A, and new Theorem 6.6. 5.Typos corrected.This is the version to appear in Journal of Statistical Physic

Convergence of random zeros on complex manifolds

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N, orthonormalized on a regular compact subset K of C^m, almost surely converge to the equilibrium measure on K as the degree N goes to infinity.Comment: 16 page

Correlations between zeros of a random polynomial

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval $(-1,1)$ are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random polynomial.Comment: 31 pages, 2 figures; a revised version of the J. Stat. Phys. pape

Infinitely Many Stochastically Stable Attractors

Let f be a diffeomorphism of a compact finite dimensional boundaryless manifold M exhibiting infinitely many coexisting attractors. Assume that each attractor supports a stochastically stable probability measure and that the union of the basins of attraction of each attractor covers Lebesgue almost all points of M. We prove that the time averages of almost all orbits under random perturbations are given by a finite number of probability measures. Moreover these probability measures are close to the probability measures supported by the attractors when the perturbations are close to the original map f.Comment: 14 pages, 2 figure

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos corrected; improvements on the statements and comments suggested by a referee. Keywords: singular flows, singular-hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm la