arXiv subjects

Joachim M. Buhmann

Publications and source records attributed to Joachim M. Buhmann.

48 records · Page 3Linked to original sources

arXiv2016

Scalable Adaptive Stochastic Optimization Using Random Projections

Adaptive stochastic gradient methods such as AdaGrad have gained popularity in particular for training deep neural networks. The most commonly used and studied variant maintains a diagonal matrix approximation to second order information by accumulating past gradients which are used to tune the step size adaptively. In certain situations the full-matrix variant of AdaGrad is expected to attain better performance, however in high dimensions it is computationally impractical. We present Ada-LR and RadaGrad two computationally efficient approximations to full-matrix AdaGrad based on randomized dimensionality reduction. They are able to capture dependencies between features and achieve similar performance to full-matrix AdaGrad but at a much smaller computational cost. We show that the regret of Ada-LR is close to the regret of full-matrix AdaGrad which can have an up-to exponentially smaller dependence on the dimension than the diagonal variant. Empirically, we show that Ada-LR and RadaGrad perform similarly to full-matrix AdaGrad. On the task of training convolutional neural networks as well as recurrent neural networks, RadaGrad achieves faster convergence than diagonal AdaGrad.

stat.ML↗

arXiv2016

Mean-Field Variational Inference for Gradient Matching with Gaussian Processes

Gradient matching with Gaussian processes is a promising tool for learning parameters of ordinary differential equations (ODE's). The essence of gradient matching is to model the prior over state variables as a Gaussian process which implies that the joint distribution given the ODE's and GP kernels is also Gaussian distributed. The state-derivatives are integrated out analytically since they are modelled as latent variables. However, the state variables themselves are also latent variables because they are contaminated by noise. Previous work sampled the state variables since integrating them out is \textit{not} analytically tractable. In this paper we use mean-field approximation to establish tight variational lower bounds that decouple state variables and are therefore, in contrast to the integral over state variables, analytically tractable and even concave for a restricted family of ODE's, including nonlinear and periodic ODE's. Such variational lower bounds facilitate "hill climbing" to determine the maximum a posteriori estimate of ODE parameters. An additional advantage of our approach over sampling methods is the determination of a proxy to the intractable posterior distribution over state variables given observations and the ODE's.

stat.ML↗

arXiv2016

Model Selection for Gaussian Process Regression by Approximation Set Coding

Gaussian processes are powerful, yet analytically tractable models for supervised learning. A Gaussian process is characterized by a mean function and a covariance function (kernel), which are determined by a model selection criterion. The functions to be compared do not just differ in their parametrization but in their fundamental structure. It is often not clear which function structure to choose, for instance to decide between a squared exponential and a rational quadratic kernel. Based on the principle of approximation set coding, we develop a framework for model selection to rank kernels for Gaussian process regression. In our experiments approximation set coding shows promise to become a model selection criterion competitive with maximum evidence (also called marginal likelihood) and leave-one-out cross-validation.

stat.ML↗

arXiv2016

TI-POOLING: transformation-invariant pooling for feature learning in Convolutional Neural Networks

In this paper we present a deep neural network topology that incorporates a simple to implement transformation invariant pooling operator (TI-POOLING). This operator is able to efficiently handle prior knowledge on nuisance variations in the data, such as rotation or scale changes. Most current methods usually make use of dataset augmentation to address this issue, but this requires larger number of model parameters and more training data, and results in significantly increased training time and larger chance of under- or overfitting. The main reason for these drawbacks is that the learned model needs to capture adequate features for all the possible transformations of the input. On the other hand, we formulate features in convolutional neural networks to be transformation-invariant. We achieve that using parallel siamese architectures for the considered transformation set and applying the TI-POOLING operator on their outputs before the fully-connected layers. We show that this topology internally finds the most optimal "canonical" instance of the input image for training and therefore limits the redundancy in learned features. This more efficient use of training data results in better performance on popular benchmark datasets with smaller number of parameters when comparing to standard convolutional neural networks with dataset augmentation and to other baselines.

cs.CV↗

arXiv2016

Greedy MAXCUT Algorithms and their Information Content

MAXCUT defines a classical NP-hard problem for graph partitioning and it serves as a typical case of the symmetric non-monotone Unconstrained Submodular Maximization (USM) problem. Applications of MAXCUT are abundant in machine learning, computer vision and statistical physics. Greedy algorithms to approximately solve MAXCUT rely on greedy vertex labelling or on an edge contraction strategy. These algorithms have been studied by measuring their approximation ratios in the worst case setting but very little is known to characterize their robustness to noise contaminations of the input data in the average case. Adapting the framework of Approximation Set Coding, we present a method to exactly measure the cardinality of the algorithmic approximation sets of five greedy MAXCUT algorithms. Their information contents are explored for graph instances generated by two different noise models: the edge reversal model and Gaussian edge weights model. The results provide insights into the robustness of different greedy heuristics and techniques for MAXCUT, which can be used for algorithm design of general USM problems.

cs.DS↗

arXiv2015

Computational Pathology: Challenges and Promises for Tissue Analysis

The histological assessment of human tissue has emerged as the key challenge for detection and treatment of cancer. A plethora of different data sources ranging from tissue microarray data to gene expression, proteomics or metabolomics data provide a detailed overview of the health status of a patient. Medical doctors need to assess these information sources and they rely on data driven automatic analysis tools. Methods for classification, grouping and segmentation of heterogeneous data sources as well as regression of noisy dependencies and estimation of survival probabilities enter the processing workflow of a pathology diagnosis system at various stages. This paper reports on state-of-the-art of the design and effectiveness of computational pathology workflows and it discusses future research directions in this emergent field of medical informatics and diagnostic machine learning.

cs.CV↗

arXiv2014

Fast and Robust Least Squares Estimation in Corrupted Linear Models

Subsampling methods have been recently proposed to speed up least squares estimation in large scale settings. However, these algorithms are typically not robust to outliers or corruptions in the observed covariates. The concept of influence that was developed for regression diagnostics can be used to detect such corrupted observations as shown in this paper. This property of influence -- for which we also develop a randomized approximation -- motivates our proposed subsampling algorithm for large scale corrupted linear regression which limits the influence of data points since highly influential points contribute most to the residual error. Under a general model of corrupted observations, we show theoretically and empirically on a variety of simulated and real datasets that our algorithm improves over the current state-of-the-art approximation schemes for ordinary least squares.

stat.ML↗

arXiv2013

Correlated random features for fast semi-supervised learning

This paper presents Correlated Nystrom Views (XNV), a fast semi-supervised algorithm for regression and classification. The algorithm draws on two main ideas. First, it generates two views consisting of computationally inexpensive random features. Second, XNV applies multiview regression using Canonical Correlation Analysis (CCA) on unlabeled data to bias the regression towards useful features. It has been shown that, if the views contains accurate estimators, CCA regression can substantially reduce variance with a minimal increase in bias. Random views are justified by recent theoretical and empirical work showing that regression with random features closely approximates kernel regression, implying that random views can be expected to contain accurate estimators. We show that XNV consistently outperforms a state-of-the-art algorithm for semi-supervised learning: substantially improving predictive performance and reducing the variability of performance on a wide variety of real-world datasets, whilst also reducing runtime by orders of magnitude.

stat.ML↗

arXiv2013

Role Mining with Probabilistic Models

Role mining tackles the problem of finding a role-based access control (RBAC) configuration, given an access-control matrix assigning users to access permissions as input. Most role mining approaches work by constructing a large set of candidate roles and use a greedy selection strategy to iteratively pick a small subset such that the differences between the resulting RBAC configuration and the access control matrix are minimized. In this paper, we advocate an alternative approach that recasts role mining as an inference problem rather than a lossy compression problem. Instead of using combinatorial algorithms to minimize the number of roles needed to represent the access-control matrix, we derive probabilistic models to learn the RBAC configuration that most likely underlies the given matrix. Our models are generative in that they reflect the way that permissions are assigned to users in a given RBAC configuration. We additionally model how user-permission assignments that conflict with an RBAC configuration emerge and we investigate the influence of constraints on role hierarchies and on the number of assignments. In experiments with access-control matrices from real-world enterprises, we compare our proposed models with other role mining methods. Our results show that our probabilistic models infer roles that generalize well to new system users for a wide variety of data, while other models' generalization abilities depend on the dataset given.

cs.CR↗

arXiv2012

Learning Dictionaries with Bounded Self-Coherence

Sparse coding in learned dictionaries has been established as a successful approach for signal denoising, source separation and solving inverse problems in general. A dictionary learning method adapts an initial dictionary to a particular signal class by iteratively computing an approximate factorization of a training data matrix into a dictionary and a sparse coding matrix. The learned dictionary is characterized by two properties: the coherence of the dictionary to observations of the signal class, and the self-coherence of the dictionary atoms. A high coherence to the signal class enables the sparse coding of signal observations with a small approximation error, while a low self-coherence of the atoms guarantees atom recovery and a more rapid residual error decay rate for the sparse coding algorithm. The two goals of high signal coherence and low self-coherence are typically in conflict, therefore one seeks a trade-off between them, depending on the application. We present a dictionary learning method with an effective control over the self-coherence of the trained dictionary, enabling a trade-off between maximizing the sparsity of codings and approximating an equiangular tight frame.

stat.ML↗

arXiv2011

Selecting the rank of truncated SVD by Maximum Approximation Capacity

Truncated Singular Value Decomposition (SVD) calculates the closest rank-$k$ approximation of a given input matrix. Selecting the appropriate rank $k$ defines a critical model order choice in most applications of SVD. To obtain a principled cut-off criterion for the spectrum, we convert the underlying optimization problem into a noisy channel coding problem. The optimal approximation capacity of this channel controls the appropriate strength of regularization to suppress noise. In simulation experiments, this information theoretic method to determine the optimal rank competes with state-of-the art model selection techniques.

cs.IT↗

arXiv2010

Information theoretic model validation for clustering

Model selection in clustering requires (i) to specify a suitable clustering principle and (ii) to control the model order complexity by choosing an appropriate number of clusters depending on the noise level in the data. We advocate an information theoretic perspective where the uncertainty in the measurements quantizes the set of data partitionings and, thereby, induces uncertainty in the solution space of clusterings. A clustering model, which can tolerate a higher level of fluctuations in the measurements than alternative models, is considered to be superior provided that the clustering solution is equally informative. This tradeoff between \emph{informativeness} and \emph{robustness} is used as a model selection criterion. The requirement that data partitionings should generalize from one data set to an equally probable second data set gives rise to a new notion of structure induced information.

cs.IT↗