251 research outputs found
Submodular Load Clustering with Robust Principal Component Analysis
Traditional load analysis is facing challenges with the new electricity usage
patterns due to demand response as well as increasing deployment of distributed
generations, including photovoltaics (PV), electric vehicles (EV), and energy
storage systems (ESS). At the transmission system, despite of irregular load
behaviors at different areas, highly aggregated load shapes still share similar
characteristics. Load clustering is to discover such intrinsic patterns and
provide useful information to other load applications, such as load forecasting
and load modeling. This paper proposes an efficient submodular load clustering
method for transmission-level load areas. Robust principal component analysis
(R-PCA) firstly decomposes the annual load profiles into low-rank components
and sparse components to extract key features. A novel submodular cluster
center selection technique is then applied to determine the optimal cluster
centers through constructed similarity graph. Following the selection results,
load areas are efficiently assigned to different clusters for further load
analysis and applications. Numerical results obtained from PJM load demonstrate
the effectiveness of the proposed approach.Comment: Accepted by 2019 IEEE PES General Meeting, Atlanta, G
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New perspectives and applications for greedy algorithms in machine learning
Approximating probability densities is a core problem in Bayesian statistics, where the inference involves the computation of a posterior distribution. Variational Inference (VI) is a technique to approximate posterior distributions through optimization. It involves specifying a set of tractable densities, out of which the final approximation is to be chosen. While VI is traditionally motivated with the goal of tractability, the focus in this dissertation is to use Bayesian approximation to obtain parsimonious distributions. With this goal in mind, we develop greedy algorithm variants and study their theoretical properties by establishing novel connections of the resulting optimization problems in parsimonious VI with traditional studies in the discrete optimization literature. Specific realizations lead to efficient solutions for many sparse probabilistic models like Sparse regression, Sparse PCA, Sparse Collective Matrix Factorization (CMF) etc. For cases where existing results are insufficient to provide acceptable approximation guarantees, we extend the optimization results for some large scale algorithms to a much larger class of functions.The developed methods are applied to both simulated and real world datasets, including high dimensional functional Magnetic Resonance Imaging (fMRI) datasets, and to the real world tasks of interpreting data exploration and model predictions.Electrical and Computer Engineerin
Algorithms and Hardness for Robust Subspace Recovery
We consider a fundamental problem in unsupervised learning called
\emph{subspace recovery}: given a collection of points in ,
if many but not necessarily all of these points are contained in a
-dimensional subspace can we find it? The points contained in are
called {\em inliers} and the remaining points are {\em outliers}. This problem
has received considerable attention in computer science and in statistics. Yet
efficient algorithms from computer science are not robust to {\em adversarial}
outliers, and the estimators from robust statistics are hard to compute in high
dimensions.
Are there algorithms for subspace recovery that are both robust to outliers
and efficient? We give an algorithm that finds when it contains more than a
fraction of the points. Hence, for say this estimator
is both easy to compute and well-behaved when there are a constant fraction of
outliers. We prove that it is Small Set Expansion hard to find when the
fraction of errors is any larger, thus giving evidence that our estimator is an
{\em optimal} compromise between efficiency and robustness.
As it turns out, this basic problem has a surprising number of connections to
other areas including small set expansion, matroid theory and functional
analysis that we make use of here.Comment: Appeared in Proceedings of COLT 201
Quantum machine learning: a classical perspective
Recently, increased computational power and data availability, as well as
algorithmic advances, have led machine learning techniques to impressive
results in regression, classification, data-generation and reinforcement
learning tasks. Despite these successes, the proximity to the physical limits
of chip fabrication alongside the increasing size of datasets are motivating a
growing number of researchers to explore the possibility of harnessing the
power of quantum computation to speed-up classical machine learning algorithms.
Here we review the literature in quantum machine learning and discuss
perspectives for a mixed readership of classical machine learning and quantum
computation experts. Particular emphasis will be placed on clarifying the
limitations of quantum algorithms, how they compare with their best classical
counterparts and why quantum resources are expected to provide advantages for
learning problems. Learning in the presence of noise and certain
computationally hard problems in machine learning are identified as promising
directions for the field. Practical questions, like how to upload classical
data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde
Proximal Methods for Hierarchical Sparse Coding
Sparse coding consists in representing signals as sparse linear combinations
of atoms selected from a dictionary. We consider an extension of this framework
where the atoms are further assumed to be embedded in a tree. This is achieved
using a recently introduced tree-structured sparse regularization norm, which
has proven useful in several applications. This norm leads to regularized
problems that are difficult to optimize, and we propose in this paper efficient
algorithms for solving them. More precisely, we show that the proximal operator
associated with this norm is computable exactly via a dual approach that can be
viewed as the composition of elementary proximal operators. Our procedure has a
complexity linear, or close to linear, in the number of atoms, and allows the
use of accelerated gradient techniques to solve the tree-structured sparse
approximation problem at the same computational cost as traditional ones using
the L1-norm. Our method is efficient and scales gracefully to millions of
variables, which we illustrate in two types of applications: first, we consider
fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we
apply our optimization tools in the context of dictionary learning, where
learned dictionary elements naturally organize in a prespecified arborescent
structure, leading to a better performance in reconstruction of natural image
patches. When applied to text documents, our method learns hierarchies of
topics, thus providing a competitive alternative to probabilistic topic models
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