537 research outputs found
On Recursive Edit Distance Kernels with Application to Time Series Classification
This paper proposes some extensions to the work on kernels dedicated to
string or time series global alignment based on the aggregation of scores
obtained by local alignments. The extensions we propose allow to construct,
from classical recursive definition of elastic distances, recursive edit
distance (or time-warp) kernels that are positive definite if some sufficient
conditions are satisfied. The sufficient conditions we end-up with are original
and weaker than those proposed in earlier works, although a recursive
regularizing term is required to get the proof of the positive definiteness as
a direct consequence of the Haussler's convolution theorem. The classification
experiment we conducted on three classical time warp distances (two of which
being metrics), using Support Vector Machine classifier, leads to conclude
that, when the pairwise distance matrix obtained from the training data is
\textit{far} from definiteness, the positive definite recursive elastic kernels
outperform in general the distance substituting kernels for the classical
elastic distances we have tested.Comment: 14 page
Optimization with Sparsity-Inducing Penalties
Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. They were first dedicated to linear variable
selection but numerous extensions have now emerged such as structured sparsity
or kernel selection. It turns out that many of the related estimation problems
can be cast as convex optimization problems by regularizing the empirical risk
with appropriate non-smooth norms. The goal of this paper is to present from a
general perspective optimization tools and techniques dedicated to such
sparsity-inducing penalties. We cover proximal methods, block-coordinate
descent, reweighted -penalized techniques, working-set and homotopy
methods, as well as non-convex formulations and extensions, and provide an
extensive set of experiments to compare various algorithms from a computational
point of view
Kernel Methods for Surrogate Modeling
This chapter deals with kernel methods as a special class of techniques for
surrogate modeling. Kernel methods have proven to be efficient in machine
learning, pattern recognition and signal analysis due to their flexibility,
excellent experimental performance and elegant functional analytic background.
These data-based techniques provide so called kernel expansions, i.e., linear
combinations of kernel functions which are generated from given input-output
point samples that may be arbitrarily scattered. In particular, these
techniques are meshless, do not require or depend on a grid, hence are less
prone to the curse of dimensionality, even for high-dimensional problems.
In contrast to projection-based model reduction, we do not necessarily assume
a high-dimensional model, but a general function that models input-output
behavior within some simulation context. This could be some micro-model in a
multiscale-simulation, some submodel in a coupled system, some initialization
function for solvers, coefficient function in PDEs, etc.
First, kernel surrogates can be useful if the input-output function is
expensive to evaluate, e.g. is a result of a finite element simulation. Here,
acceleration can be obtained by sparse kernel expansions. Second, if a function
is available only via measurements or a few function evaluation samples, kernel
approximation techniques can provide function surrogates that allow global
evaluation.
We present some important kernel approximation techniques, which are kernel
interpolation, greedy kernel approximation and support vector regression.
Pseudo-code is provided for ease of reproducibility. In order to illustrate the
main features, commonalities and differences, we compare these techniques on a
real-world application. The experiments clearly indicate the enormous
acceleration potentia
Using Mean Embeddings for State Estimation and Reinforcement Learning
To act in complex, high-dimensional environments, autonomous systems require versatile state estimation techniques and compact state representations. State estimation is crucial when the system only has access to stochastic measurements or partial observations. Furthermore, in combination with models of the system such techniques allow to predict the future which enables the system to asses the outcome of possible decisions. Compact state representations alleviate the curse of dimensionality by distilling the important information from high-dimensional observations.
Due to noisy sensory information and non-perfect models of the system, estimates of the state never reflect the true state perfectly but are always subject to errors. The natural choice to incorporate the uncertainty about the state estimate is to use a probability distribution as representation. This results in the so called belief state.
High-dimensional observations, for example images, often contain much less information than conveyed by their dimensionality. But also if all the information is necessary to describe the state of the system—for example, think of the state of a swarm with the positions of all agents—a less complex description might be a sufficient representation. In such situations, finding the generative distribution that explains the state would give a much more compact while informative representation.
Traditionally, parametric distributions have been used as state representations such as most prevalently the Gaussian distribution. However, in many cases a unimodal distribution might not be sufficient to represent the belief state. Using multi-modal probability distributions, instead, requires more advanced approaches such as mixture models or particle-based Monte Carlo methods. Learning mixture models is however not straight-forward and often results in locally optimal solutions. Similarly, maintaining a good population of particles during inference is a complicated and cumbersome process. A third approach is kernel density estimation which is located at the intersection of mixture models and particle-based approaches. Still, performing inference with any of these approaches requires heuristics that lead to poor performance and a limited scalability to higher dimensional spaces.
A recent technique that alleviates this problem are the embeddings of probability distributions into reproducing kernel Hilbert spaces (RKHS). Conditional distributions can be embedded as operators based on which a framework for inference has been presented that allows to apply the sum rule, the product rule and Bayes’ rule entirely in Hilbert space. Using sample based estimators and the kernel-trick of the representer theorem allows to represent the operations as vector-matrix manipulations. The contributions of this thesis are based on or inspired by the embeddings of distributions into reproducing kernel Hilbert spaces.
In the first part of this thesis, I propose additions to the framework for nonparametric inference that allow the inference operators to scale more gracefully with the number of samples in the training set. The first contribution is an alternative approach to the conditional embedding operator formulated as a least-squares problem
i which allows to use only a subset of the data as representation while using the full data set to learn the conditional operator. I call this operator the subspace conditional embedding operator. Inspired by the least-squares derivations of the Kalman filter, I furthermore propose an alternative operator for Bayesian updates in Hilbert space, the kernel Kalman rule. This alternative approach is numerically more robust than the kernel Bayes rule presented in the framework for non-parametric inference and scales better with the number of samples. Based on the kernel Kalman rule, I derive the kernel Kalman filter and the kernel forward-backward smoother to perform state estimation, prediction and smoothing based on Hilbert space embeddings of the belief state. This representation is able to capture multi-modal distributions and inference resolves--due to the kernel trick--into easy matrix manipulations.
In the second part of this thesis, I propose a representation for large sets of homogeneous observations. Specifically, I consider the problem of learning a controller for object assembly and object manipulation with a robotic swarm. I assume a swarm of homogeneous robots that are controlled by a common input signal, e.g., the gradient of a light source or a magnetic field. Learning policies for swarms is a challenging problem since the state space grows with the number of agents and becomes quickly very high dimensional. Furthermore, the exact number of agents and the order of the agents in the observation is not important to solve the task. To approach this issue, I propose the swarm kernel which uses a Hilbert space embedding to represent the swarm. Instead of the exact positions of the agents in the swarm, the embedding estimates the generative distribution behind the swarm configuration. The specific agent positions are regarded as samples of this distribution. Since the swarm kernel compares the embeddings of distributions, it can compare swarm configurations with varying numbers of individuals and is invariant to the permutation of the agents. I present a hierarchical approach for solving the object manipulation task where I assume a high-level object assembly policy as given. To learn the low-level object pushing policy, I use the swarm kernel with an actor-critic policy search method. The policies which I learn in simulation can be directly transferred to a real robotic system.
In the last part of this thesis, I investigate how we can employ the idea of kernel mean embeddings to deep reinforcement learning. As in the previous part, I consider a variable number of homogeneous observations—such as robot swarms where the number of agents can change. Another example is the representation of 3D structures as point clouds. The number of points in such clouds can vary strongly and the order of the points in a vectorized representation is arbitrary. The common architectures for neural networks have a fixed structure that requires that the dimensionality of inputs and outputs is known in advance. A variable number of inputs can only be processed by applying tricks. To approach this problem, I propose the deep M-embeddings which are inspired by the kernel mean embeddings. The deep M-embeddings provide a network structure to compute a fixed length representation from a variable number of inputs. Additionally, the deep M-embeddings exploit the homogeneous nature of the inputs to reduce the number of parameters in the network and, thus, make the learning easier. Similar to the swarm kernel, the policies learned with the deep M-embeddings can be transferred to different swarm sizes and different number of objects in the environment without further learning
Convolutional Sparse Kernel Network for Unsupervised Medical Image Analysis
The availability of large-scale annotated image datasets and recent advances
in supervised deep learning methods enable the end-to-end derivation of
representative image features that can impact a variety of image analysis
problems. Such supervised approaches, however, are difficult to implement in
the medical domain where large volumes of labelled data are difficult to obtain
due to the complexity of manual annotation and inter- and intra-observer
variability in label assignment. We propose a new convolutional sparse kernel
network (CSKN), which is a hierarchical unsupervised feature learning framework
that addresses the challenge of learning representative visual features in
medical image analysis domains where there is a lack of annotated training
data. Our framework has three contributions: (i) We extend kernel learning to
identify and represent invariant features across image sub-patches in an
unsupervised manner. (ii) We initialise our kernel learning with a layer-wise
pre-training scheme that leverages the sparsity inherent in medical images to
extract initial discriminative features. (iii) We adapt a multi-scale spatial
pyramid pooling (SPP) framework to capture subtle geometric differences between
learned visual features. We evaluated our framework in medical image retrieval
and classification on three public datasets. Our results show that our CSKN had
better accuracy when compared to other conventional unsupervised methods and
comparable accuracy to methods that used state-of-the-art supervised
convolutional neural networks (CNNs). Our findings indicate that our
unsupervised CSKN provides an opportunity to leverage unannotated big data in
medical imaging repositories.Comment: Accepted by Medical Image Analysis (with a new title 'Convolutional
Sparse Kernel Network for Unsupervised Medical Image Analysis'). The
manuscript is available from following link
(https://doi.org/10.1016/j.media.2019.06.005
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Advances in Probabilistic Meta-Learning and the Neural Process Family
A natural progression in machine learning research is to automate and learn from data increasingly many components of our learning agents.Meta-learning is a paradigm that fully embraces this perspective, and can be intuitively described as embodying the idea of learning to learn. A goal of meta-learning research is the development of models to assist users in navigating the intricate space of design choices associated with specifying machine learning solutions. This space is particularly formidable when considering deep learning approaches, which involve myriad design choices interacting in complex fashions to affect the performance of the resulting agents. Despite the impressive successes of deep learning in recent years, this challenge remains a significant bottleneck in deploying neural network based solutions in several important application domains. But how can we reason about and design solutions to this daunting task?
This thesis is concerned with a particular perspective for meta-learning in supervised settings. We view supervised learning algorithms as mappings that take data sets to predictive models, and consider meta-learning as learning to approximate functions of this form. In particular, we are interested in meta-learners that (i) employ neural networks to approximate these functions in an end-to-end manner, and (ii) provide predictive distributions rather than single predictors. The former is motivated by the success of neural networks as function approximators, and the latter by our interest in the few-shot learning scenario. The introductory chapters of this thesis formalise this notion, and use it to provide a tutorial introducing the Neural Process Family (NPF), a class of models introduced by Garnelo et al (2018) satisfying the above-mentioned modelling desiderata. We then present our own technical contributions to the NPF.
First, we focus on fundamental properties of the model-class, such as expressivity and limiting behaviours of the associated training procedures. Next, we study the role of translation equivariance in the NPF. Considering the intimate relationship between the NPF and the representation of functions operating on sets, we extend the underlying theory of DeepSets to include translation equivariance. We then develop novel members of the NPF endowed with this important inductive bias. Through extensive empirical evaluation, we demonstrate that, in many settings, they significantly outperform their non-equivariant counterparts.
Finally, we turn our attention to the development of Neural Processes for few-shot image-classification. We introduce models that navigate the important tradeoffs associated with this setting, and describe the specification of their central components. We demonstrate that the resulting models---CNAPs---achieve state-of-the-art performance on a challenging benchmark called Meta-Dataset, while adapting faster and with less computational overhead than their best-performing competitors
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