58 research outputs found
Omnidirectional Bats, Point-to-Plane Distances, and the Price of Uniqueness
We study simultaneous localization and mapping with a device that uses
reflections to measure its distance from walls. Such a device can be realized
acoustically with a synchronized collocated source and receiver; it behaves
like a bat with no capacity for directional hearing or vocalizing. In this
paper we generalize our previous work in 2D, and show that the 3D case is not
just a simple extension, but rather a fundamentally different inverse problem.
While generically the 2D problem has a unique solution, in 3D uniqueness is
always absent in rooms with fewer than nine walls. In addition to the complete
characterization of ambiguities which arise due to this non-uniqueness, we
propose a robust solution for inexact measurements similar to analogous results
for Euclidean Distance Matrices. Our theoretical results have important
consequences for the design of collocated range-only SLAM systems, and we
support them with an array of computer experiments.Comment: 5 pages, 8 figures, submitted to ICASSP 201
Scalable and Robust Community Detection with Randomized Sketching
This paper explores and analyzes the unsupervised clustering of large
partially observed graphs. We propose a scalable and provable randomized
framework for clustering graphs generated from the stochastic block model. The
clustering is first applied to a sub-matrix of the graph's adjacency matrix
associated with a reduced graph sketch constructed using random sampling. Then,
the clusters of the full graph are inferred based on the clusters extracted
from the sketch using a correlation-based retrieval step. Uniform random node
sampling is shown to improve the computational complexity over clustering of
the full graph when the cluster sizes are balanced. A new random degree-based
node sampling algorithm is presented which significantly improves upon the
performance of the clustering algorithm even when clusters are unbalanced. This
algorithm improves the phase transitions for matrix-decomposition-based
clustering with regard to computational complexity and minimum cluster size,
which are shown to be nearly dimension-free in the low inter-cluster
connectivity regime. A third sampling technique is shown to improve balance by
randomly sampling nodes based on spatial distribution. We provide analysis and
numerical results using a convex clustering algorithm based on matrix
completion
The achievable performance of convex demixing
Demixing is the problem of identifying multiple structured signals from a
superimposed, undersampled, and noisy observation. This work analyzes a general
framework, based on convex optimization, for solving demixing problems. When
the constituent signals follow a generic incoherence model, this analysis leads
to precise recovery guarantees. These results admit an attractive
interpretation: each signal possesses an intrinsic degrees-of-freedom
parameter, and demixing can succeed if and only if the dimension of the
observation exceeds the total degrees of freedom present in the observation
Information Recovery from Pairwise Measurements
A variety of information processing tasks in practice involve recovering
objects from single-shot graph-based measurements, particularly those taken
over the edges of some measurement graph . This paper concerns the
situation where each object takes value over a group of different values,
and where one is interested to recover all these values based on observations
of certain pairwise relations over . The imperfection of
measurements presents two major challenges for information recovery: 1)
: a (dominant) portion of measurements are
corrupted; 2) : a significant fraction of pairs are
unobservable, i.e. can be highly sparse.
Under a natural random outlier model, we characterize the , that is, the critical threshold of non-corruption rate
below which exact information recovery is infeasible. This accommodates a very
general class of pairwise relations. For various homogeneous random graph
models (e.g. Erdos Renyi random graphs, random geometric graphs, small world
graphs), the minimax recovery rate depends almost exclusively on the edge
sparsity of the measurement graph irrespective of other graphical
metrics. This fundamental limit decays with the group size at a square root
rate before entering a connectivity-limited regime. Under the Erdos Renyi
random graph, a tractable combinatorial algorithm is proposed to approach the
limit for large (), while order-optimal recovery is
enabled by semidefinite programs in the small regime.
The extended (and most updated) version of this work can be found at
(http://arxiv.org/abs/1504.01369).Comment: This version is no longer updated -- please find the latest version
at (arXiv:1504.01369
Convexity in source separation: Models, geometry, and algorithms
Source separation or demixing is the process of extracting multiple
components entangled within a signal. Contemporary signal processing presents a
host of difficult source separation problems, from interference cancellation to
background subtraction, blind deconvolution, and even dictionary learning.
Despite the recent progress in each of these applications, advances in
high-throughput sensor technology place demixing algorithms under pressure to
accommodate extremely high-dimensional signals, separate an ever larger number
of sources, and cope with more sophisticated signal and mixing models. These
difficulties are exacerbated by the need for real-time action in automated
decision-making systems.
Recent advances in convex optimization provide a simple framework for
efficiently solving numerous difficult demixing problems. This article provides
an overview of the emerging field, explains the theory that governs the
underlying procedures, and surveys algorithms that solve them efficiently. We
aim to equip practitioners with a toolkit for constructing their own demixing
algorithms that work, as well as concrete intuition for why they work
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