the subspace information criterion for infinite dimensional hypothesis spaces a central problem in learning is selection of an appropriate model this is typically done by estimating the unknown generalization errors of a set of models to be selected from and then choosing the model with minimal generalization error estimate in this article we discuss the problem of model selection and generalization error estimation in the context of kernel regression models e g kernel ridge regression kernel subset regression or gaussian process regression previously a non asymptotic generalization error estimator called the subspace information criterion sic was proposed that could be successfully applied to finite dimensional subspace models sic is an unbiased estimator of the generalization error for the finite sample case under the conditions that the learning target function belongs to a specified reproducing kernel hilbert space rkhs h and the reproducing kernels centered on training sample points span the whole space h these conditions hold only if dim h l where l is the number of training examples therefore sic could be applied only to finite dimensional rkhss in this paper we extend the range of applicability of sic and show that even if the reproducing kernels centered on training sample points do not span the whole space h sic is an unbiased estimator of an essential part of the generalization error our extension allows the use of any rkhss including infinite dimensional ones i e richer function classes commonly used in gaussian processes support vector machines or boosting we further show that when the kernel matrix is invertible sic can be expressed in a much simpler form making its computation highly efficient in computer simulations on ridge parameter selection with real and artificial data sets sic is compared favorably with other standard model selection techniques for instance leave one out cross validation or an empirical bayesian method

active learning applied to rcs computations with nonuniform sampling using different objective functions an active learning framework is introduced to reduce the number of frequencies and angles one must consider for wideband monostatic scattering computations or measurements this method is used to optimally select those frequencies and angles that would be most informative resulting in nonuniform sampling and often a reduced number of points vis agrave vis uniform sampling in this paper we focus on jointly two dimensional optimal sampling in frequency and incident angle thetas for monostatic scattering the method consists of two basic steps one step involves estimation of model parameters using a least square ls algorithm the next step is to optimally choose the next point frequency and thetas for analysis by the computational model or experiment this new point is selected with the goal of reducing uncertainty in the parametric model quantified via the fisher information matrix iterating these two steps a sequence of numerical computations or measurements are performed each at the most informative point for learning the parameters of the associated simpler parametric model this idea is demonstrated here in the context of reducing the number of points frequencies and orientations at which a computational model must be employed and in order to avoid repeatedly gathering samplings at the edge of the input space an alternative objective function is applied which makes the actively selected points closer to a region of interest

inferring solutions of differential equations using noisy multi fidelity data for more than two centuries solutions of differential equations have been obtained either analytically or numerically based on typically well behaved forcing and boundary conditions for well posed problems we are changing this paradigm in a fundamental way by establishing an interface between probabilistic machine learning and differential equations we develop data driven algorithms for general linear equations using gaussian process priors tailored to the corresponding integro differential operators the only observables are scarce noisy multi fidelity data for the forcing and solution that are not required to reside on the domain boundary the resulting predictive posterior distributions quantify uncertainty and naturally lead to adaptive solution refinement via active learning this general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time integration and is scalable to high dimensions

learning unrealizable tasks from minimum entropy queries in supervised learning learning from queries rather than from random examples can improve generalization performance significantly we study the performance of query learning for unrealizable tasks where the student cannot learn from the perfectly as a simple model scenario of this kind we consider a linear perceptron student learning a general nonlinear perceptron teacher two kinds of queries for maximum information gain i e minimum entropy are investigated minimum student space entropy msse queries which are appropriate if the teacher space is unknown and minimum teacher space entropy mtse queries which can be used if the teacher space is assumed to be known but a student of a simpler form has deliberately been chosen we find that for msse queries the structure of the student space determines the efficacy of query learning mtse queries on the other hand which we investigate for the extreme case of a binary perceptron teacher lead to a higher generalization error than random examples due to a lack of feedback about the progress of the student in the way queries are selected

incremental active learning for optimal generalization the problem of designing input signals for optimal generalization is called active learning in this article we give a two stage sampling scheme for reducing both the bias and variance and based on this scheme we propose two active learning methods one is the multipoint search method applicable to arbitrary models the effectiveness of this method is shown through computer simulations the other is the optimal sampling method in trigonometric polynomial models this method precisely specifies the optimal sampling locations

near optimal sensor placement for linear inverse problems a classic problem is the estimation of a set of parameters from measurements collected by only a few sensors the number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates unfortunately the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest we propose framesense a greedy algorithm for the selection of optimal sensor locations the core cost function of the algorithm is the frame potential a scalar property of matrices that measures the orthogonality of its rows notably framesense is the first algorithm that is near optimal in terms of mean square error meaning that its solution is always guaranteed to be close to the optimal one moreover we show with an extensive set of numerical experiments that framesense achieves state of the art performance while having the lowest computational cost when compared to other greedy methods

generalized spike and slab priors for bayesian group feature selection using expectation propagation we describe a bayesian method for group feature selection in linear regression problems the method is based on a generalized version of the standard spike and slab prior distribution which is often used for individual feature selection exact bayesian inference under the prior considered is infeasible for typical regression problems however approximate inference can be carried out efficiently using expectation propagation ep a detailed analysis of the generalized spike and slab prior shows that it is well suited for regression problems that are sparse at the group level furthermore this prior can be used to introduce prior knowledge about specific groups of features that are a priori believed to be more relevant an experimental evaluation compares the performance of the proposed method with those of group lasso bayesian group lasso automatic relevance determination and additional variants used for group feature selection the results of these experiments show that a model based on the generalized spike and slab prior and the ep algorithm has state of the art prediction performance in the problems analyzed furthermore this model is also very useful to carry out sequential experimental design also known as active learning where the data instances that are most informative are iteratively included in the training set reducing the number of instances needed to obtain a particular level of prediction accuracy

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

adaptive design optimization a mutual information based approach to model discrimination in cognitive science discriminating among competing statistical models is a pressing issue for many experimentalists in the field of cognitive science resolving this issue begins with designing maximally informative experiments to this end the problem to be solved in adaptive design optimization is identifying experimental designs under which one can infer the underlying model in the fewest possible steps when the models under consideration are nonlinear as is often the case in cognitive science this problem can be impossible to solve analytically without simplifying assumptions however as we show in this letter a full solution can be found numerically with the help of a bayesian computational trick derived from the statistics literature which recasts the problem as a probability density simulation in which the optimal design is the mode of the density we use a utility function based on mutual information and give three intuitive interpretations of the utility function in terms of bayesian posterior estimates as a proof of concept we offer a simple example application to an experiment on memory retention

theory of genetic algorithms ii models for genetic operators over the string tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling we present a theoretical framework for an asymptotically converging scaled genetic algorithm which uses an arbitrary size alphabet and common scaled genetic operators the alphabet can be interpreted as a set of equidistant real numbers and multiple spot mutation performs a scalable compromise between pure random search and neighborhood based change on the alphabet level we discuss several versions of the crossover operator and their interplay with mutation in particular we consider uniform crossover and gene lottery crossover which does not commute with mutation the vose liepins version of mutation crossover is also integrated in our approach in order to achieve convergence to global optima the mutation rate and the crossover rate have to be annealed to zero in proper fashion and unbounded power law scaled proportional fitness selection is used with logarithmic growth in the exponent our analysis shows that using certain types of crossover operators and large population size allows for particularly slow annealing schedules for the crossover rate in our discussion we focus on the following three major aspects based upon contraction properties of the mutation and fitness selection operators i the drive towards uniform populations in a genetic algorithm using standard operations ii weak ergodicity of the inhomogeneous markov chain describing the probabilistic model for the scaled algorithm iii convergence to globally optimal solutions in particular we remove two restrictions imposed in theorem 8 6 and remark 8 7 of theoret comput sci 259 2001 1 where a similar type of algorithm is considered as described here mutation need not commute with crossover and the fitness function which may come from a coevolutionary single species setting need not have a single maximum

an information theoretic approach of designing sparse kernel adaptive filters this paper discusses an information theoretic approach of designing sparse kernel adaptive filters to determine useful data to be learned and remove redundant ones a subjective information measure called surprise is introduced surprise captures the amount of information a datum contains which is transferable to a learning system based on this concept we propose a systematic sparsification scheme which can drastically reduce the time and space complexity without harming the performance of kernel adaptive filters nonlinear regression short term chaotic time series prediction and long term time series forecasting examples are presented

how to modify a neural network gradually without changing its input output functionality it is generally unknown when distinct neural networks having different synaptic weights and thresholds implement identical input output transformations determining the exact conditions for structurally distinct yet functionally equivalent networks may shed light on the theoretical constraints on how diverse neural circuits might develop and be maintained to serve identical functions such consideration also imposes practical limits on our ability to uniquely infer the structure of underlying neural circuits from stimulus response measurements we introduce a biologically inspired mathematical method for determining when the structure of a neural network can be perturbed gradually while preserving functionality we show that for common three layer networks with convergent and nondegenerate connection weights this is possible only when the hidden unit gains are power functions exponentials or logarithmic functions which are known to approximate the gains seen in some biological neurons for practical applications our numerical simulations with finite and noisy data show that continuous confounding of parameters due to network functional equivalence tends to occur approximately even when the gain function is not one of the aforementioned three types suggesting that our analytical results are applicable to more general situations and may help identify a common source of parameter variability in neural network modeling

active learning in approximately linear regression based on conditional expectation of generalization error the goal of active learning is to determine the locations of training input points so that the generalization error is minimized we discuss the problem of active learning in linear regression scenarios traditional active learning methods using least squares learning often assume that the model used for learning is correctly specified in many practical situations however this assumption may not be fulfilled recently active learning methods using importance weighted least squares learning have been proposed which are shown to be robust against misspecification of models in this paper we propose a new active learning method also using the weighted least squares learning which we call alice active learning using the importance weighted least squares learning based on conditional expectation of the generalization error an important difference from existing methods is that we predict the conditional expectation of the generalization error given training input points while existing methods predict the full expectation of the generalization error due to this difference the training input design can be fine tuned depending on the realization of training input points theoretically we prove that the proposed active learning criterion is a more accurate predictor of the single trial generalization error than the existing criterion numerical studies with toy and benchmark data sets show that the proposed method compares favorably to existing methods

appendix to the iwasawa conjecture for totally real fields by a wiles arithmetic minimal compactification of the hilbert blumenthal moduli spaces the arithmetic theory of toroidal compactification of the hilbert blumenthal moduli spaces was described in rapoport s thesis rap while the arithmetic theory of minimal compactification has been lacking for a long time after faltings work on the arithmetic compactification of siegel moduli spaces fa it was obvious to the experts that the same things could be done in the hilbertblumenthal case in similar but simpler ways in fact what emerged was a machinery to compactify moduli spaces coming from abelian varieties it turns out that the arithmetic minimal compactification of the hilbert blumenthal moduli spaces could have been done more than ten years ago rap contains everything that is needed so in some sense i am writing a summary and an appendix to rap the purpose of this appendix is to explain this remark and show that this theory of minimal compactification can be useful in number theory for instance one can replace for almost all primes by for all primes outside the level and the discriminant in the main result of br la two more applications are given here one about p adic monodromy the other about positivity of the hodge line bundle the sections of which are hilbert modular forms for a systematic exposition of the theory of semi abelian degeneration and compactification the reader is referred to the book fa ch where the more

a stochastic self organizing map for proximity data we derive an efficient algorithm for topographic mapping of proximity data tmp which can be seen as an extension of kohonen s self organizing map to arbitrary distance measures the tmp cost function is derived in a baysian framework of folded markov chains for the description of autoencoders it incorporates the data by a dissimilarity matrix and the topographic neighborhood by a matrix of transition probabilities from the principle of maximum entropy a nonfactorizing gibbs distribution is obtained which is approximated in a mean field fashion this allows for maximum likelihood estimation using an expectation maximization algorithm in analogy to the transition from topographic vector quantization to the self organizing map we suggest an approximation to tmp that is computationally more efficient in order to prevent convergence to local minima an annealing scheme in the temperature parameter is introduced for which the critical temperature of the first phase transition is calculated in terms o

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

selecting optimal experiments for multiple output multilayer perceptrons where should a researcher conduct experiments to provide training data for a multilayer perceptron this question is investigated and a statistical method for selecting optimal experimental design points for multiple output multilayer perceptrons is introduced multiple class discrimination problems are examined using a framework in which the multilayer perceptron is viewed as a multivariate nonlinear regression model following a bayesian formulation for the case where the variance covariance matrix of the responses is unknown a selection criterion is developed this criterion is based on the volume of the joint confidence ellipsoid for the weights in a multilayer perceptron an example is used to demonstrate the superiority of optimally selected design points over randomly chosen points as well as points chosen in a grid pattern simplification of the basic criterion is offered through the use of hadamard matrices to produce uncorrelated outputs

rankin eisenstein classes for modular forms in this paper we make a systematic study of certain motivic cohomology classes rankin eisenstein classes attached to the rankin selberg convolution of two modular forms of weight 2 the main result is the computation of the p adic syntomic regulators of these classes as a consequence we prove many cases of the perrin riou conjecture for rankin selberg convolutions of cusp forms

constructive mathematics a foundation for computable analysis this paper introduces bishop s constructive mathematics which can be regarded as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable analysis the real numbers are presented using a set of constructive axioms from which are derived some elementary properties of the real line r including its completeness

multi output separable gaussian process towards an efficient fully bayesian paradigm for uncertainty quantification computer codes simulating physical systems usually have responses that consist of a set of distinct outputs e g velocity and pressure that evolve also in space and time and depend on many unknown input parameters e g physical constants initial boundary conditions etc furthermore essential engineering procedures such as uncertainty quantification inverse problems or design are notoriously difficult to carry out mostly due to the limited simulations available the aim of this work is to introduce a fully bayesian approach for treating these problems which accounts for the uncertainty induced by the finite number of observations our model is built on a multi dimensional gaussian process that explicitly treats correlations between distinct output variables as well as space and or time the proper use of a separable covariance function enables us to describe the huge covariance matrix as a kronecker product of smaller matrices leading to efficient algorithms for carrying out inference and predictions the novelty of this work is the recognition that the gaussian process model defines a posterior probability measure on the function space of possible surrogates for the computer code and the derivation of an algorithmic procedure that allows us to sample it efficiently we demonstrate how the scheme can be used in uncertainty quantification tasks in order to obtain error bars for the statistics of interest that account for the finite number of observations

characterizing responses of translation invariant neurons to natural stimuli maximally informative invariant dimensions the human visual system is capable of recognizing complex objects even when their appearances change drastically under various viewing conditions especially in the higher cortical areas the sensory neurons reflect such functional capacity in their selectivity for complex visual features and invariance to certain object transformations such as image translation due to the strong nonlinearities necessary to achieve both the selectivity and invariance characterizing and predicting the response patterns of these neurons represents a formidable computational challenge a related problem is that such neurons are poorly driven by randomized inputs such as white noise and respond strongly only to stimuli with complex high order correlations such as natural stimuli here we describe a novel two step optimization technique that can characterize both the shape selectivity and the range and coarseness of position invariance from neural responses to natural stimuli one step in the optimization is finding the template as the maximally informative dimension given the estimated spatial location where the response could have been triggered within each image the estimates of the locations that triggered the response are updated in the next step under the assumption of a monotonic relationship between the firing rate and stimulus projections on the template at a given position the most likely location is the one that has the largest projection on the estimate of the template the algorithm shows quick convergence during optimization and the estimation results are reliable even in the regime of small signal to noise ratios when we apply the algorithm to responses of complex cells in the primary visual cortex v1 to natural movies we find that responses of the majority of cells were significantly better described by translation invariant models based on one template compared with position specific models with several relevant features

a practical bayesian framework for backpropagation networks a quantitative and practical bayesian framework is described for learning of mappings in feedforward networks the framework makes possible 1 objective comparisons between solutions using alternative network architectures 2 objective stopping rules for network pruning or growing procedures 3 objective choice of magnitude and type of weight decay terms or additive regularizers for penalizing large weights etc 4 a measure of the effective number of well determined parameters in a model 5 quantified estimates of the error bars on network parameters and on network output and 6 objective comparisons with alternative learning and interpolation models such as splines and radial basis functions the bayesian evidence automatically embodies occam s razor penalizing overflexible and overcomplex models the bayesian approach helps detect poor underlying assumptions in learning models for learning models well matched to a problem a good correlation between generalization ability and the bayesian evidence is obtained

sensor placement by maximal projection on minimum eigenspace for linear inverse problems this paper presents two new greedy sensor placement algorithms named minimum nonzero eigenvalue pursuit mnep and maximal projection on minimum eigenspace mpme for linear inverse problems with greater emphasis on the mpme algorithm for performance comparison with existing approaches in both mnep and mpme we select the sensing locations one by one in this way the least number of required sensor nodes can be determined by checking whether the estimation accuracy is satisfied after each sensing location is determined for the mpme algorithm the minimum eigenspace is defined as the eigenspace associated with the minimum eigenvalue of the dual observation matrix for each sensing location the projection of its observation vector onto the minimum eigenspace is shown to be monotonically decreasing w r t the worst case error variance wcev of the estimated parameters we select the sensing location whose observation vector has the maximum projection onto the minimum eigenspace of the current dual observation matrix the proposed mpme is shown to be one of the most computationally efficient algorithms our monte carlo simulations showed that mpme outperforms the convex relaxation method the sparsense method and the framesense method in terms of wcev and the mean square error mse of the estimated parameters especially when the number of available sensor nodes is very limited

sensor centric data reduction for estimation with wsns via censoring and quantization consider a wireless sensor network wsn with a fusion center fc deployed to estimate signal parameters from noisy sensor measurements if the wsn has a large number of low cost battery operated sensor nodes with limited transmission bandwidth then conservation of transmission resources power and bandwidth is paramount to this end the present paper develops a novel data reduction method which requires no inter sensor collaboration and results in only a subset of the sensor measurements transmitted to the fc using interval censoring as a data reduction method each sensor decides separately whether to censor its acquired measurements based on a rule that promotes censoring of measurements with least impact on the estimator mean square error mse leveraging the statistical distribution of sensor data the censoring mechanism and the received uncensored data fc based estimators are derived for both deterministic via maximum likelihood estimation and random parameters via maximum a posteriori probability estimation for a linear gaussian model quantization of the uncensored measurements at the sensor nodes offers an additional degree of freedom in the resource conservation versus estimator mse reduction tradeoff cramer rao bound analysis for the different censor estimators and censor quantizer estimators is also provided to benchmark and facilitate mse based performance comparisons numerical simulations corroborate the analytical findings and demonstrate that the proposed censoring estimation approach performs competitively with alternative methods under different sensing conditions while having lower computational complexity

an information based learning approach to dual control dual control aims to concurrently learn and control an unknown system however actively learning the system conflicts directly with any given control objective for it will disturb the system during exploration this paper presents a receding horizon approach to dual control where a multiobjective optimization problem is solved repeatedly and subject to constraints representing system dynamics balancing a standard finite horizon control objective a knowledge gain objective is defined to explicitly quantify the information acquired when learning the system dynamics measures from information theory such as entropy based uncertainty fisher information and relative entropy are studied and used to quantify the knowledge gained as a result of the control actions the resulting iterative framework is applied to markov decision processes and discrete time nonlinear systems thus the broad applicability and usefulness of the presented approach is demonstrated in diverse problem settings the framework is illustrated with multiple numerical examples

asymptotic theory of information theoretic experimental design we discuss an idea for collecting data in a relatively efficient manner our point of view is bayesian and information theoretic on any given trial we want to adaptively choose the input in such a way that the mutual information between the unknown state of the system and the stochastic output is maximal given any prior information including data collected on any previous trials we prove a theorem that quantifies the effectiveness of this strategy and give a few illustrative examples comparing the performance of this adaptive technique to that of the more usual nonadaptive experimental design in particular we calculate the asymptotic efficiency of the information maximization strategy and demonstrate that this method is in a well defined sense never less efficient and is generically more efficient than the nonadaptive strategy for example we are able to explicitly calculate the asymptotic relative efficiency of the staircase method widely employed in psychophysics research and to demonstrate the dependence of this efficiency on the form of the psychometric function underlying the output responses

parallelizing exploration exploitation tradeoffs in gaussian process bandit optimization how can we take advantage of opportunities for experimental parallelization in exploration exploitation tradeoffs in many experimental scenarios it is often desirable to execute experiments simultaneously or in batches rather than only performing one at a time additionally observations may be both noisy and expensive we introduce gaussian process batch upper confidence bound gp bucb an upper confidence bound based algorithm which models the reward function as a sample from a gaussian process and which can select batches of experiments to run in parallel we prove a general regret bound for gp bucb as well as the surprising result that for some common kernels the asymptotic average regret can be made independent of the batch size the gp bucb algorithm is also applicable in the related case of a delay between initiation of an experiment and observation of its results for which the same regret bounds hold we also introduce gaussian process adaptive upper confidence bound gp aucb a variant of gp bucb which can exploit parallelism in an adaptive manner we evaluate gp bucb and gp aucb on several simulated and real data sets these experiments show that gp bucb and gp aucb are competitive with state of the art heuristics

active state estimation for nonlinear systems a neural approximation approach in this paper we consider the problem of actively providing an estimate of the state of a stochastic dynamic system over a possibly long finite time horizon the active estimation problem aep is formulated as a stochastic optimal control one in which the minimization of a suitable uncertainty measure is carried out toward this end the use of the renyi entropy as an information measure is proposed and motivated a neural control scheme based on the application of the extended ritz method erim and on the use of a gaussian sum filter gsf is then presented simulation results show the effectiveness of the proposed approach

minimax bounds for active learning this paper analyzes the potential advantages and theoretical challenges of active learning algorithms active learning involves sequential sampling procedures that use information gleaned from previous samples in order to focus the sampling and accelerate the learning process relative to passive learning algorithms which are based on nonadaptive usually random samples there are a number of empirical and theoretical results suggesting that in certain situations active learning can be significantly more effective than passive learning however the fact that active learning algorithms are feedback systems makes their theoretical analysis very challenging this paper aims to shed light on achievable limits in active learning using minimax analysis techniques we study the achievable rates of classification error convergence for broad classes of distributions characterized by decision boundary regularity and noise conditions the results clearly indicate the conditions under which one can expect significant gains through active learning furthermore we show that the learning rates derived are tight for boundary fragment classes in d dimensional feature spaces when the feature marginal density is bounded from above and below

active learning and basis selection for kernel based linear models a bayesian perspective we develop an active learning algorithm for kernel based linear regression and classification the proposed greedy algorithm employs a minimum entropy criterion derived using a bayesian interpretation of ridge regression we assume access to a matrix bbrn n for which the i j th element is defined by the kernel function k i j bbr with the observed data i bbrd we seek a model m i yi where yi is a real valued response or integer valued label which we do not have access to a priori to achieve this goal a submatrix il ib bbrn m is sought that corresponds to the intersection of n rows and m columns of indexed by the sets il and ib respectively typically m n and n n we have two objectives i determine the m columns of indexed by the set ib that are the most informative for building a linear model m 1 i ib t yi without any knowledge of yi i 1n and ii using active learning sequentially determine which subset of n elements of yi i 1n should be acquired both stopping values ib m and il n are also to be inferred from the data these steps are taken with the goal of minimizing the uncertainty of the model parameters x as measured by the differential entropy of its posterior distribution the parameter vector x bbrm as well as the model bias bbr is then learned from the resulting problem yil il ibx 1 the remaining n n responses labels not included in yil can be inferred by applying x to the remaining n n rows of ib we show experimental results for several regression and classification problems and compare to other active learning methods

variational bayes for continuous hidden markov models and its application to active learning in this paper we present a variational bayes vb framework for learning continuous hidden markov models chmms and we examine the vb framework within active learning unlike a maximum likelihood or maximum a posteriori training procedure which yield a point estimate of the chmm parameters vb based training yields an estimate of the full posterior of the model parameters this is particularly important for small training sets since it gives a measure of confidence in the accuracy of the learned model this is utilized within the context of active learning for which we acquire labels for those feature vectors for which knowledge of the associated label would be most informative for reducing model parameter uncertainty three active learning algorithms are considered in this paper 1 query by committee qbc with the goal of selecting data for labeling that minimize the classification variance 2 a maximum expected information gain method that seeks to label data with the goal of reducing the entropy of the model parameters and 3 an error reduction based procedure that attempts to minimize classification error over the test data the experimental results are presented for synthetic and measured data we demonstrate that all of these active learning methods can significantly reduce the amount of required labeling compared to random selection of samples for labeling

an interactive approach to solving correspondence problems finding correspondences among objects in different images is a critical problem in computer vision even good correspondence procedures can fail however when faced with deformations occlusions and differences in lighting and zoom levels across images we present a methodology for augmenting correspondence matching algorithms with a means for triaging the focus of attention and effort in assisting the automated matching for guiding the mix of human and automated initiatives we introduce a measure of the expected value of resolving correspondence uncertainties we explore the value of the approach with experiments on benchmark data

sequential optimal design of neurophysiology experiments adaptively optimizing experiments has the potential to significantly reduce the number of trials needed to build parametric statistical models of neural systems however application of adaptive methods to neurophysiology has been limited by severe computational challenges since most neurons are high dimensional systems optimizing neurophysiology experiments requires computing high dimensional integrations and optimizations in real time here we present a fast algorithm for choosing the most informative stimulus by maximizing the mutual information between the data and the unknown parameters of a generalized linear model glm that we want to fit to the neuron s activity we rely on important log concavity and asymptotic normality properties of the posterior to facilitate the required computations our algorithm requires only low rank matrix manipulations and a two dimensional search to choose the optimal stimulus the average running time of these operations scales quadratically with the dimensionality of the glm making real time adaptive experimental design feasible even for high dimensional stimulus and parameter spaces for example we require roughly 10 milliseconds on a desktop computer to optimize a 100 dimensional stimulus despite using some approximations to make the algorithm efficient our algorithm asymptotically decreases the uncertainty about the model parameters at a rate equal to the maximum rate predicted by an asymptotic analysis simulation results show that picking stimuli by maximizing the mutual information can speed up convergence to the optimal values of the parameters by an order of magnitude compared to using random nonadaptive stimuli finally applying our design procedure to real neurophysiology experiments requires addressing the nonstationarities that we would expect to see in neural responses our algorithm can efficiently handle both fast adaptation due to spike history effects and slow nonsystematic drifts in a neuron s activity

machine learning of linear differential equations using gaussian processes this work leverages recent advances in probabilistic machine learning to discover governing equations expressed by parametric linear operators such equations involve but are not limited to ordinary and partial differential integro differential and fractional order operators here gaussian process priors are modified according to the particular form of such operators and are employed to infer parameters of the linear equations from scarce and possibly noisy observations such observations may come from experiments or black box computer simulations as demonstrated in several synthetic examples and a realistic application in functional genomics employ probabilistic machine learning to discover governing equations expressed by parametric linear operators proper placement of gaussian process priors allows one to efficiently infer model parameters via maximum likelihood estimation a general treatment of inverse problems governed by linear operators leading to model discovery from just a handful of noisy measurements

variational inference in nonconjugate models mean field variational methods are widely used for approximate posterior inference in many probabilistic models in a typical application mean field methods approximately compute the posterior with a coordinate ascent optimization algorithm when the model is conditionally conjugate the coordinate updates are easily derived and in closed form however many models of interest like the correlated topic model and bayesian logistic regression are nonconjugate in these models mean field methods cannot be directly applied and practitioners have had to develop variational algorithms on a case by case basis in this paper we develop two generic methods for nonconjugate models laplace variational inference and delta method variational inference our methods have several advantages they allow for easily derived variational algorithms with a wide class of nonconjugate models they extend and unify some of the existing algorithms that have been derived for specific models and they work well on real world data sets we studied our methods on the correlated topic model bayesian logistic regression and hierarchical bayesian logistic regression

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

gaussian processes for object categorization discriminative methods for visual object category recognition are typically non probabilistic predicting class labels but not directly providing an estimate of uncertainty gaussian processes gps provide a framework for deriving regression techniques with explicit uncertainty models we show here how gaussian processes with covariance functions defined based on a pyramid match kernel pmk can be used for probabilistic object category recognition our probabilistic formulation provides a principled way to learn hyperparameters which we utilize to learn an optimal combination of multiple covariance functions it also offers confidence estimates at test points and naturally allows for an active learning paradigm in which points are optimally selected for interactive labeling we show that with an appropriate combination of kernels a significant boost in classification performance is possible further our experiments indicate the utility of active learning with probabilistic predictive models especially when the amount of training data labels that may be sought for a category is ultimately very small

a certified corecursive implementation of exact real numbers we implement exact real numbers in the logical framework coq using streams i e infinite sequences of digits and characterize constructive real numbers through a minimal axiomatization we prove that our construction inhabits the axiomatization working formally with coinductive types and corecursive proofs thus we obtain reliable corecursive algorithms for computing on real numbers

sequential design of computer experiments for the assessment of fetus exposure to electromagnetic fields in this article we describe four sequential sampling strategies for estimating the quantile of y f x where x has a known distribution in rd and f is a deterministic unknown expensive to evaluate real valued function these approaches all consist in modeling f as a sample of a well chosen gaussian process and aim at estimating the quantile by using as few evaluations of f as possible the different methodologies are first compared through various numerical experiments then in the framework of the anr jst fetus project we apply our strategies to a real example corresponding to the exposure of a japanese pregnant woman model and her 26 week old fetus to a plane wave finally we compare our methodologies on a simplified geometric model designed for modeling the fetus exposure to plane waves supplementary materials for this article are available online

covariate shift adaptation by importance weighted cross validation a common assumption in supervised learning is that the input points in the training set follow the same probability distribution as the input points that will be given in the future test phase however this assumption is not satisfied for example when the outside of the training region is extrapolated the situation where the training input points and test input points follow different distributions while the conditional distribution of output values given input points is unchanged is called the covariate shift under the covariate shift standard model selection techniques such as cross validation do not work as desired since its unbiasedness is no longer maintained in this paper we propose a new method called importance weighted cross validation iwcv for which we prove its unbiasedness even under the covariate shift the iwcv procedure is the only one that can be applied for unbiased classification under covariate shift whereas alternatives to iwcv exist for regression the usefulness of our proposed method is illustrated by simulations and furthermore demonstrated in the brain computer interface where strong non stationarity effects can be seen between training and test sessions

bayesian computation using design of experiments based interpolation technique in this article a new deterministic approximation method for bayesian computation known as design of experiments based interpolation technique doit is proposed the method works by sampling points from the parameter space using an experimental design and then fitting a kriging model to interpolate the unnormalized posterior the approximated posterior density is a weighted average of normal densities and therefore most of the posterior quantities can be easily computed doit is a general computing technique that is easy to implement and can be applied to many complex bayesian problems moreover it does not suffer from the curse of dimensionality as much as some quadrature methods it can work using fewer posterior evaluations which is a great advantage over the monte carlo and markov chain monte carlo methods especially when dealing with computationally expensive posteriors this article has supplementary material that is available online

information rates of nonparametric gaussian process methods we consider the quality of learning a response function by a nonparametric bayesian approach using a gaussian process gp prior on the response function we upper bound the quadratic risk of the learning procedure which in turn is an upper bound on the kullback leibler information between the predictive and true data distribution the upper bound is expressed in small ball probabilities and concentration measures of the gp prior we illustrate the computation of the upper bound for the matern and squared exponential kernels for these priors the risk and hence the information criterion tends to zero for all continuous response functions however the rate at which this happens depends on the combination of true response function and gaussian prior and is expressible in a certain concentration function in particular the results show that for good performance the regularity of the gp prior should match the regularity of the unknown response function

the evidence framework applied to classification networks three bayesian ideas are presented for supervised adaptive classifiers first it is argued that the output of a classifier should be obtained by marginalizing over the posterior distribution of the parameters a simple approximation to this integral is proposed and demonstrated this involves a moderation of the most probable classifier s outputs and yields improved performance second it is demonstrated that the bayesian framework for model comparison described for regression models in mackay 1992a b can also be applied to classification problems this framework successfully chooses the magnitude of weight decay terms and ranks solutions found using different numbers of hidden units third an information based data selection criterion is derived and demonstrated within this framework

multi output local gaussian process regression applications to uncertainty quantification we develop an efficient bayesian uncertainty quantification framework using a novel treed gaussian process model the tree is adaptively constructed using information conveyed by the observed data about the length scales of the underlying process on each leaf of the tree we utilize bayesian experimental design techniques in order to learn a multi output gaussian process the constructed surrogate can provide analytical point estimates as well as error bars for the statistics of interest we numerically demonstrate the effectiveness of the suggested framework in identifying discontinuities local features and unimportant dimensions in the solution of stochastic differential equations

adaptive hessian based nonstationary gaussian process response surface method for probability density approximation with application to bayesian solution of large scale inverse problems we develop an adaptive hessian based non stationary gaussian process gp response surface method for approximating a probability density function pdf that exploits its structure particularly the hessian of its negative logarithm of particular interest to us are pdfs that arise from the bayesian solution of large scale inverse problems which imply very expensive to evaluate pdfs the method can be considered as a piecewise adaptive gaussian approximation in which a gaussian tailored to the local hessian of the negative log probability density is constructed for each subregion in high dimensional parameter space the task of efficiently partitioning the parameter space into subregions is done implicitly through hessian informed membership probability functions the gp machinery is then employed to glue all local gaussian approximations into a global analytical response surface that is far cheaper to evaluate than the original expensive probability density the resulting response surface is also equipp

modularity of the rankin selberg l series and multiplicity one for sl 2 let f g be primitive cusp forms holomorphic or otherwise on the upper half plane h of levels n m respectively with unitarily normalized l functions l s f equation and l s g equation when p does not divide n resp m the inverse roots p p resp p p are nonzero with sum ap resp bp for every p prime to nm set lp s f g 1 p pp s 1 p pp s 1 p pp s 1 p pp s 1 let l s f g denote the incomplete euler product of lp s f g over all p not dividing nm this is closely related to the convolution l series sum over n 1 a sub n b sub n n s whose miraculous properties were first studied by rankin and selberg

bayesian active learning of neural firing rate maps with transformed gaussian process priors a firing rate map also known as a tuning curve describes the nonlinear relationship between a neuron s spike rate and a low dimensional stimulus e g orientation head direction contrast color here we investigate bayesian active learning methods for estimating firing rate maps in closed loop neurophysiology experiments these methods can accelerate the characterization of such maps through the intelligent adaptive selection of stimuli specifically we explore the manner in which the prior and utility function used in bayesian active learning affect stimulus selection and performance our approach relies on a flexible model that involves a nonlinearly transformed gaussian process gp prior over maps and conditionally poisson spiking we show that infomax learning which selects stimuli to maximize the information gain about the firing rate map exhibits strong dependence on the seemingly innocuous choice of nonlinear transformation function we derive an alternate utility function that selects stimuli to minimize the average posterior variance of the firing rate map and analyze the surprising relationship between prior parameterization stimulus selection and active learning performance in gp poisson models we apply these methods to color tuning measurements of neurons in macaque primary visual cortex

entropy search for information efficient global optimization contemporary global optimization algorithms are based on local measures of utility rather than a probability measure over location and value of the optimum they thus attempt to collect low function values not to learn about the optimum the reason for the absence of probabilistic global optimizers is that the corresponding inference problem is intractable in several ways this paper develops desiderata for probabilistic optimization algorithms then presents a concrete algorithm which addresses each of the computational intractabilities with a sequence of approximations and explicitly addresses the decision problem of maximizing information gain from each evaluation

dethroning the fano factor a flexible model based approach to partitioning neural variability neurons in many brain areas exhibit high trial to trial variability with spike counts that are overdispersed relative to a poisson distribution recent work goris movshon simoncelli 2014 ha

dynamic trees for learning and design dynamic regression trees are an attractive option for automatic regression and classification with complicated response surfaces in online application settings we create a sequential tree model whose state changes in time with the accumulation of new data and provide particle learning algorithms that allow for the efficient online posterior filtering of tree states a major advantage of tree regression is that it allows for the use of very simple models within each partition the model also facilitates a natural division of labor in our sequential particle based inference tree dynamics are defined through a few potential changes that are local to each newly arrived observation while global uncertainty is captured by the ensemble of particles we consider both constant and linear mean functions at the tree leaves along with multinomial leaves for classification problems and propose default prior specifications that allow for prediction to be integrated over all model parameters conditional on a given

on the chow motive of an abelian scheme with non trivial endomorphisms let x be an abelian scheme over a base variety s with endomorphism algebra d we prove that the relative chow motive r x s has a canonical decomposition as a direct sum of motives r whereruns over an explicitly determined finite set of irreducible representations of the group d op such that r seen as a functor from chow motives to d op representations is isotypic our decomposition refines the motivic decomposition of deninger and murre as well as beauville s decomposition of the chow group the second main result is that we construct a canonical generalized motivic lefschetz decomposition inspired by work of looijenga and lunts in 17 the role of sl2 in the classical theory is here replaced by a larger lie algebra generated by all lefschetz and lambda operators associated to non degenerate line bundles we construct an action of this larger lie algebra on r x s and deduce from this a generalized lefschetz decomposition we also give a precise structure result for the lefschetz components that arise as an application of our techniques we provide a positive answer to a question of claire voisin concerning the analogue for abelian varieties of a conjecture of beauville ams 2010 mathematics subject classification 14c15 14k05