48,814 research outputs found
Raiders of the Lost Architecture: Kernels for Bayesian Optimization in Conditional Parameter Spaces
In practical Bayesian optimization, we must often search over structures with
differing numbers of parameters. For instance, we may wish to search over
neural network architectures with an unknown number of layers. To relate
performance data gathered for different architectures, we define a new kernel
for conditional parameter spaces that explicitly includes information about
which parameters are relevant in a given structure. We show that this kernel
improves model quality and Bayesian optimization results over several simpler
baseline kernels.Comment: 6 pages, 3 figures. Appeared in the NIPS 2013 workshop on Bayesian
optimizatio
RG flows of Quantum Einstein Gravity on maximally symmetric spaces
We use the Wetterich-equation to study the renormalization group flow of
-gravity in a three-dimensional, conformally reduced setting. Building on
the exact heat kernel for maximally symmetric spaces, we obtain a partial
differential equation which captures the scale-dependence of for
positive and, for the first time, negative scalar curvature. The effects of
different background topologies are studied in detail and it is shown that they
affect the gravitational RG flow in a way that is not visible in
finite-dimensional truncations. Thus, while featuring local background
independence, the functional renormalization group equation is sensitive to the
topological properties of the background. The detailed analytical and numerical
analysis of the partial differential equation reveals two globally well-defined
fixed functionals with at most a finite number of relevant deformations. Their
properties are remarkably similar to two of the fixed points identified within
the -truncation of full Quantum Einstein Gravity. As a byproduct, we
obtain a nice illustration of how the functional renormalization group realizes
the "integrating out" of fluctuation modes on the three-sphere.Comment: 35 pages, 6 figure
Large-scale Nonlinear Variable Selection via Kernel Random Features
We propose a new method for input variable selection in nonlinear regression.
The method is embedded into a kernel regression machine that can model general
nonlinear functions, not being a priori limited to additive models. This is the
first kernel-based variable selection method applicable to large datasets. It
sidesteps the typical poor scaling properties of kernel methods by mapping the
inputs into a relatively low-dimensional space of random features. The
algorithm discovers the variables relevant for the regression task together
with learning the prediction model through learning the appropriate nonlinear
random feature maps. We demonstrate the outstanding performance of our method
on a set of large-scale synthetic and real datasets.Comment: Final version for proceedings of ECML/PKDD 201
Parsimonious Mahalanobis Kernel for the Classification of High Dimensional Data
The classification of high dimensional data with kernel methods is considered
in this article. Exploit- ing the emptiness property of high dimensional
spaces, a kernel based on the Mahalanobis distance is proposed. The computation
of the Mahalanobis distance requires the inversion of a covariance matrix. In
high dimensional spaces, the estimated covariance matrix is ill-conditioned and
its inversion is unstable or impossible. Using a parsimonious statistical
model, namely the High Dimensional Discriminant Analysis model, the specific
signal and noise subspaces are estimated for each considered class making the
inverse of the class specific covariance matrix explicit and stable, leading to
the definition of a parsimonious Mahalanobis kernel. A SVM based framework is
used for selecting the hyperparameters of the parsimonious Mahalanobis kernel
by optimizing the so-called radius-margin bound. Experimental results on three
high dimensional data sets show that the proposed kernel is suitable for
classifying high dimensional data, providing better classification accuracies
than the conventional Gaussian kernel
Kernel Multivariate Analysis Framework for Supervised Subspace Learning: A Tutorial on Linear and Kernel Multivariate Methods
Feature extraction and dimensionality reduction are important tasks in many
fields of science dealing with signal processing and analysis. The relevance of
these techniques is increasing as current sensory devices are developed with
ever higher resolution, and problems involving multimodal data sources become
more common. A plethora of feature extraction methods are available in the
literature collectively grouped under the field of Multivariate Analysis (MVA).
This paper provides a uniform treatment of several methods: Principal Component
Analysis (PCA), Partial Least Squares (PLS), Canonical Correlation Analysis
(CCA) and Orthonormalized PLS (OPLS), as well as their non-linear extensions
derived by means of the theory of reproducing kernel Hilbert spaces. We also
review their connections to other methods for classification and statistical
dependence estimation, and introduce some recent developments to deal with the
extreme cases of large-scale and low-sized problems. To illustrate the wide
applicability of these methods in both classification and regression problems,
we analyze their performance in a benchmark of publicly available data sets,
and pay special attention to specific real applications involving audio
processing for music genre prediction and hyperspectral satellite images for
Earth and climate monitoring
Kernel Spectral Curvature Clustering (KSCC)
Multi-manifold modeling is increasingly used in segmentation and data
representation tasks in computer vision and related fields. While the general
problem, modeling data by mixtures of manifolds, is very challenging, several
approaches exist for modeling data by mixtures of affine subspaces (which is
often referred to as hybrid linear modeling). We translate some important
instances of multi-manifold modeling to hybrid linear modeling in embedded
spaces, without explicitly performing the embedding but applying the kernel
trick. The resulting algorithm, Kernel Spectral Curvature Clustering, uses
kernels at two levels - both as an implicit embedding method to linearize
nonflat manifolds and as a principled method to convert a multiway affinity
problem into a spectral clustering one. We demonstrate the effectiveness of the
method by comparing it with other state-of-the-art methods on both synthetic
data and a real-world problem of segmenting multiple motions from two
perspective camera views.Comment: accepted to 2009 ICCV Workshop on Dynamical Visio
A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization
We present a general approach for collaborative filtering (CF) using spectral
regularization to learn linear operators from "users" to the "objects" they
rate. Recent low-rank type matrix completion approaches to CF are shown to be
special cases. However, unlike existing regularization based CF methods, our
approach can be used to also incorporate information such as attributes of the
users or the objects -- a limitation of existing regularization based CF
methods. We then provide novel representer theorems that we use to develop new
estimation methods. We provide learning algorithms based on low-rank
decompositions, and test them on a standard CF dataset. The experiments
indicate the advantages of generalizing the existing regularization based CF
methods to incorporate related information about users and objects. Finally, we
show that certain multi-task learning methods can be also seen as special cases
of our proposed approach
Conditional Similarity Networks
What makes images similar? To measure the similarity between images, they are
typically embedded in a feature-vector space, in which their distance preserve
the relative dissimilarity. However, when learning such similarity embeddings
the simplifying assumption is commonly made that images are only compared to
one unique measure of similarity. A main reason for this is that contradicting
notions of similarities cannot be captured in a single space. To address this
shortcoming, we propose Conditional Similarity Networks (CSNs) that learn
embeddings differentiated into semantically distinct subspaces that capture the
different notions of similarities. CSNs jointly learn a disentangled embedding
where features for different similarities are encoded in separate dimensions as
well as masks that select and reweight relevant dimensions to induce a subspace
that encodes a specific similarity notion. We show that our approach learns
interpretable image representations with visually relevant semantic subspaces.
Further, when evaluating on triplet questions from multiple similarity notions
our model even outperforms the accuracy obtained by training individual
specialized networks for each notion separately.Comment: CVPR 201
The Sample Complexity of Dictionary Learning
A large set of signals can sometimes be described sparsely using a
dictionary, that is, every element can be represented as a linear combination
of few elements from the dictionary. Algorithms for various signal processing
applications, including classification, denoising and signal separation, learn
a dictionary from a set of signals to be represented. Can we expect that the
representation found by such a dictionary for a previously unseen example from
the same source will have L_2 error of the same magnitude as those for the
given examples? We assume signals are generated from a fixed distribution, and
study this questions from a statistical learning theory perspective.
We develop generalization bounds on the quality of the learned dictionary for
two types of constraints on the coefficient selection, as measured by the
expected L_2 error in representation when the dictionary is used. For the case
of l_1 regularized coefficient selection we provide a generalization bound of
the order of O(sqrt(np log(m lambda)/m)), where n is the dimension, p is the
number of elements in the dictionary, lambda is a bound on the l_1 norm of the
coefficient vector and m is the number of samples, which complements existing
results. For the case of representing a new signal as a combination of at most
k dictionary elements, we provide a bound of the order O(sqrt(np log(m k)/m))
under an assumption on the level of orthogonality of the dictionary (low Babel
function). We further show that this assumption holds for most dictionaries in
high dimensions in a strong probabilistic sense. Our results further yield fast
rates of order 1/m as opposed to 1/sqrt(m) using localized Rademacher
complexity. We provide similar results in a general setting using kernels with
weak smoothness requirements
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