We also find hyperparameters of the covariance function in the same joint optimization.

There has been a surge of interest in learning nonlinear manifold models to approximate high-dimensional data. Both for computational complexity reasonsand for generalization capability, sparsity is a desired feature in such models. This usually means dimensionality reduction, which naturally implies estimating the intrinsic dimension, but it can also mean selecting a subset of the data to use as landmarks, which is especially important becausemany existing algorithms have quadratic complexity in the number of observations.

We analyze classification error on unseen cases, i.e. cases that are different fromthose in the training set. Unlike standard generalization error, this off-training-set error may differ significantly from the empirical error withhigh probability even with large sample sizes. We derive a datadependent boundon the difference between off-training-set and standard generalization error. Our result is based on a new bound on the missing mass, which for small samples is stronger than existing bounds based on Good-Turing estimators. As we demonstrate on UCI data-sets, our bound gives nontrivial generalization guarantees in many practical cases. In light of these results, we show that certain claims made in the No Free Lunch literature are overly pessimistic.

The problem of computing a resample estimate for the reconstruction error in PCA is reformulated as an inference problem with the help of the replica method. Using the expectation consistent (EC) approximation, theintractable inference problem can be solved efficiently using only two variational parameters. A perturbative correction to the result is computed and an alternative simplified derivation is also presented.

When trying to understand the brain, it is of fundamental importance to analyse (e.g. from EEG/MEG measurements) what parts of the cortex interact with each other in order to infer more accurate models of brain activity. Common techniques like Blind Source Separation (BSS) can estimate brainsources and single out artifacts by using the underlying assumption ofsource signal independence. However, physiologically interesting brain sources typically interact, so BSS will--by construction-- fail to characterize them properly. Noting that there are truly interacting sources and signals that only seemingly interact due to effects of volume conduction, this work aims to contribute by distinguishing these effects. For this a new BSS technique is proposed that uses anti-symmetrized cross-correlation matrices and subsequent diagonalization. The resulting decomposition consists of the truly interacting brain sources and suppresses anyspurious interaction stemming from volume conduction. Our new concept of interacting source analysis (ISA) is successfully demonstrated onMEG data.

This paper provides a system-level analysis of a scalable distributed sensing modelfor networked sensors. In our system model, a data center acquires datafrom a bunch of L sensors which each independently encode their noisy observations of an original binary sequence, and transmit their encoded data sequences to the data center at a combined rate R, which is limited. Supposing that the sensors use independent LDGM rate distortion codes,we show that the system performance can be evaluated for any given finite R when the number of sensors L goes to infinity. The analysis shows how the optimal strategy for the distributed sensing problem changesat critical values of the data rate R or the noise level.

Sparse PCA seeks approximate sparse "eigenvectors" whose projections capture the maximal variance of data. As a cardinality-constrained and non-convex optimization problem, it is NPhard and is encountered in a wide range of applied fields, from bio-informatics to finance. Recent progress has focused mainly on continuous approximation and convex relaxation of the hard cardinality constraint. In contrast, we consider an alternative discrete spectral formulation based on variational eigenvalue bounds and provide an effective greedy strategy as well as provably optimal solutions using branch-and-bound search. Moreover, the exact methodology used reveals a simple renormalization step that improves approximate solutions obtained by any continuous method. The resulting performance gain of discrete algorithms is demonstrated on real-world benchmark data and in extensive Monte Carlo evaluation trials.

We considered a gamma distribution of interspike intervals as a statistical modelfor neuronal spike generation. The model parameters consist of a time-dependent firing rate and a shape parameter that characterizes spiking irregularities of individual neurons. Because the environment changes with time, observed data are generated from the time-dependent firing rate, which is an unknown function. A statistical model with an unknown function is called a semiparametric model, which is one of the unsolved problem in statistics and is generally very difficult to solve. We used a novel method of estimating functions in information geometry to estimate the shape parameter without estimating the unknown function. We analytically obtained an optimal estimating function for the shape parameter independent of the functional form of the firing rate. This estimation is efficient without Fisher information loss and better than maximum likelihood estimation.

The network topology of neurons in the brain exhibits an abundance of feedback connections, but the computational function of these feedback connections is largely unknown. We present a computational theory that characterizes the gain in computational power achieved through feedback in dynamical systems with fading memory. It implies that many such systems acquire through feedback universal computational capabilities for analog computing with a non-fading memory. In particular, we show that feedback enables such systems to process time-varying input streams in diverse ways according to rules that are implemented through internal states of the dynamical system. In contrast to previous attractor-based computational models for neural networks, these flexible internal states are high-dimensional attractors of the circuit dynamics, that still allow the circuit state to absorb new information from online input streams. In this way one arrives at novel models for working memory, integration of evidence, and reward expectation in cortical circuits. We show that they are applicable to circuits of conductance-based Hodgkin-Huxley (HH) neurons with high levels of noise that reflect experimental data on invivo conditions.

There is experimental evidence that cortical neurons show avalanche activity withthe intensity of firing events being distributed as a power-law. We present a biologically plausible extension of a neural network which exhibits a power-law avalanche distribution for a wide range of connectivity parameters.