108 research outputs found
Determining Critical Points of Handwritten Mathematical Symbols Represented as Parametric Curves
We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of hand-written mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid ill-conditioned basis conversions. Our main contribution is to assemble the relevant mathematical tools to perform all the necessary operations in the orthogonal polynomial basis. These include implicitization, differentiation, root finding and resultant computation
Sample Complexity of the Robust LQG Regulator with Coprime Factors Uncertainty
This paper addresses the end-to-end sample complexity bound for learning the
H2 optimal controller (the Linear Quadratic Gaussian (LQG) problem) with
unknown dynamics, for potentially unstable Linear Time Invariant (LTI) systems.
The robust LQG synthesis procedure is performed by considering bounded additive
model uncertainty on the coprime factors of the plant. The closed-loop
identification of the nominal model of the true plant is performed by
constructing a Hankel-like matrix from a single time-series of noisy finite
length input-output data, using the ordinary least squares algorithm from
Sarkar et al. (2020). Next, an H-infinity bound on the estimated model error is
provided and the robust controller is designed via convex optimization, much in
the spirit of Boczar et al. (2018) and Zheng et al. (2020a), while allowing for
bounded additive uncertainty on the coprime factors of the model. Our
conclusions are consistent with previous results on learning the LQG and LQR
controllers.Comment: Minor Edits on closed loop identification, 30 pages, 2 figures, 3
algorithm
Structured Sparsity Models for Multiparty Speech Recovery from Reverberant Recordings
We tackle the multi-party speech recovery problem through modeling the
acoustic of the reverberant chambers. Our approach exploits structured sparsity
models to perform room modeling and speech recovery. We propose a scheme for
characterizing the room acoustic from the unknown competing speech sources
relying on localization of the early images of the speakers by sparse
approximation of the spatial spectra of the virtual sources in a free-space
model. The images are then clustered exploiting the low-rank structure of the
spectro-temporal components belonging to each source. This enables us to
identify the early support of the room impulse response function and its unique
map to the room geometry. To further tackle the ambiguity of the reflection
ratios, we propose a novel formulation of the reverberation model and estimate
the absorption coefficients through a convex optimization exploiting joint
sparsity model formulated upon spatio-spectral sparsity of concurrent speech
representation. The acoustic parameters are then incorporated for separating
individual speech signals through either structured sparse recovery or inverse
filtering the acoustic channels. The experiments conducted on real data
recordings demonstrate the effectiveness of the proposed approach for
multi-party speech recovery and recognition.Comment: 31 page
Linear Control Theory with an ℋ∞ Optimality Criterion
This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods
Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix
We reduce the problem of computing the rank and a nullspace basis of a
univariate polynomial matrix to polynomial matrix multiplication. For an input
n x n matrix of degree d over a field K we give a rank and nullspace algorithm
using about the same number of operations as for multiplying two matrices of
dimension n and degree d. If the latter multiplication is done in
MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix
multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K.
The softO notation indicates some missing logarithmic factors. The method is
randomized with Las Vegas certification. We achieve our results in part through
a combination of matrix Hensel high-order lifting and matrix minimal fraction
reconstruction, and through the computation of minimal or small degree vectors
in the nullspace seen as a K[x]-moduleComment: Research Report LIP RR2005-03, January 200
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