168 research outputs found
Computing Bi-Lipschitz Outlier Embeddings into the Line
The problem of computing a bi-Lipschitz embedding of a graphical metric into
the line with minimum distortion has received a lot of attention. The
best-known approximation algorithm computes an embedding with distortion
, where denotes the optimal distortion [B\u{a}doiu \etal~2005]. We
present a bi-criteria approximation algorithm that extends the above results to
the setting of \emph{outliers}.
Specifically, we say that a metric space admits a
-embedding if there exists , with , such that
admits an embedding into the line with distortion at
most . Given , and a metric space that admits a -embedding,
for some , our algorithm computes a -embedding in polynomial time. This is the first algorithmic result for
outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier
embeddings where known only for the case of additive distortion
Composition of nested embeddings with an application to outlier removal
We study the design of embeddings into Euclidean space with outliers. Given a
metric space and an integer , the goal is to embed all but
points in (called the "outliers") into with the smallest possible
distortion . Finding the optimal distortion for a given outlier set size
, or alternately the smallest for a given target distortion are both
NP-hard problems. In fact, it is UGC-hard to approximate to within a factor
smaller than even when the metric sans outliers is isometrically embeddable
into . We consider bi-criteria approximations. Our main result is a
polynomial time algorithm that approximates the outlier set size to within an
factor and the distortion to within a constant factor.
The main technical component in our result is an approach for constructing a
composition of two given embeddings from subsets of into which
inherits the distortions of each to within small multiplicative factors.
Specifically, given a low distortion embedding from into
and a high(er) distortion embedding from the entire set into
, we construct a single embedding that achieves the same distortion
over pairs of points in and an expansion of at most over the remaining pairs of points, where . Our
composition theorem extends to embeddings into arbitrary metrics for
, and may be of independent interest. While unions of embeddings over
disjoint sets have been studied previously, to our knowledge, this is the first
work to consider compositions of nested embeddings.Comment: 25 pages (including 2 appendices), 5 figure
Sparse Control of Alignment Models in High Dimension
For high dimensional particle systems, governed by smooth nonlinearities
depending on mutual distances between particles, one can construct
low-dimensional representations of the dynamical system, which allow the
learning of nearly optimal control strategies in high dimension with
overwhelming confidence. In this paper we present an instance of this general
statement tailored to the sparse control of models of consensus emergence in
high dimension, projected to lower dimensions by means of random linear maps.
We show that one can steer, nearly optimally and with high probability, a
high-dimensional alignment model to consensus by acting at each switching time
on one agent of the system only, with a control rule chosen essentially
exclusively according to information gathered from a randomly drawn
low-dimensional representation of the control system.Comment: 39 page
Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms
We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques
Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph
Data-sensitive metrics adapt distances locally based the density of data
points with the goal of aligning distances and some notion of similarity. In
this paper, we give the first exact algorithm for computing a data-sensitive
metric called the nearest neighbor metric. In fact, we prove the surprising
result that a previously published -approximation is an exact algorithm.
The nearest neighbor metric can be viewed as a special case of a
density-based distance used in machine learning, or it can be seen as an
example of a manifold metric. Previous computational research on such metrics
despaired of computing exact distances on account of the apparent difficulty of
minimizing over all continuous paths between a pair of points. We leverage the
exact computation of the nearest neighbor metric to compute sparse spanners and
persistent homology. We also explore the behavior of the metric built from
point sets drawn from an underlying distribution and consider the more general
case of inputs that are finite collections of path-connected compact sets.
The main results connect several classical theories such as the conformal
change of Riemannian metrics, the theory of positive definite functions of
Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop
novel proof techniques based on the combination of screw functions and
Lipschitz extensions that may be of independent interest.Comment: 15 page
One-class classifiers based on entropic spanning graphs
One-class classifiers offer valuable tools to assess the presence of outliers
in data. In this paper, we propose a design methodology for one-class
classifiers based on entropic spanning graphs. Our approach takes into account
the possibility to process also non-numeric data by means of an embedding
procedure. The spanning graph is learned on the embedded input data and the
outcoming partition of vertices defines the classifier. The final partition is
derived by exploiting a criterion based on mutual information minimization.
Here, we compute the mutual information by using a convenient formulation
provided in terms of the -Jensen difference. Once training is
completed, in order to associate a confidence level with the classifier
decision, a graph-based fuzzy model is constructed. The fuzzification process
is based only on topological information of the vertices of the entropic
spanning graph. As such, the proposed one-class classifier is suitable also for
data characterized by complex geometric structures. We provide experiments on
well-known benchmarks containing both feature vectors and labeled graphs. In
addition, we apply the method to the protein solubility recognition problem by
considering several representations for the input samples. Experimental results
demonstrate the effectiveness and versatility of the proposed method with
respect to other state-of-the-art approaches.Comment: Extended and revised version of the paper "One-Class Classification
Through Mutual Information Minimization" presented at the 2016 IEEE IJCNN,
Vancouver, Canad
Graph Priors, Optimal Transport, and Deep Learning in Biomedical Discovery
Recent advances in biomedical data collection allows the collection of massive datasets measuring thousands of features in thousands to millions of individual cells. This data has the potential to advance our understanding of biological mechanisms at a previously impossible resolution. However, there are few methods to understand data of this scale and type. While neural networks have made tremendous progress on supervised learning problems, there is still much work to be done in making them useful for discovery in data with more difficult to represent supervision. The flexibility and expressiveness of neural networks is sometimes a hindrance in these less supervised domains, as is the case when extracting knowledge from biomedical data. One type of prior knowledge that is more common in biological data comes in the form of geometric constraints. In this thesis, we aim to leverage this geometric knowledge to create scalable and interpretable models to understand this data. Encoding geometric priors into neural network and graph models allows us to characterize the models’ solutions as they relate to the fields of graph signal processing and optimal transport. These links allow us to understand and interpret this datatype. We divide this work into three sections. The first borrows concepts from graph signal processing to construct more interpretable and performant neural networks by constraining and structuring the architecture. The second borrows from the theory of optimal transport to perform anomaly detection and trajectory inference efficiently and with theoretical guarantees. The third examines how to compare distributions over an underlying manifold, which can be used to understand how different perturbations or conditions relate. For this we design an efficient approximation of optimal transport based on diffusion over a joint cell graph. Together, these works utilize our prior understanding of the data geometry to create more useful models of the data. We apply these methods to molecular graphs, images, single-cell sequencing, and health record data
Phase Retrieval From Binary Measurements
We consider the problem of signal reconstruction from quadratic measurements
that are encoded as +1 or -1 depending on whether they exceed a predetermined
positive threshold or not. Binary measurements are fast to acquire and
inexpensive in terms of hardware. We formulate the problem of signal
reconstruction using a consistency criterion, wherein one seeks to find a
signal that is in agreement with the measurements. To enforce consistency, we
construct a convex cost using a one-sided quadratic penalty and minimize it
using an iterative accelerated projected gradient-descent (APGD) technique. The
PGD scheme reduces the cost function in each iteration, whereas incorporating
momentum into PGD, notwithstanding the lack of such a descent property,
exhibits faster convergence than PGD empirically. We refer to the resulting
algorithm as binary phase retrieval (BPR). Considering additive white noise
contamination prior to quantization, we also derive the Cramer-Rao Bound (CRB)
for the binary encoding model. Experimental results demonstrate that the BPR
algorithm yields a signal-to- reconstruction error ratio (SRER) of
approximately 25 dB in the absence of noise. In the presence of noise prior to
quantization, the SRER is within 2 to 3 dB of the CRB
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