12 research outputs found
Wavefield recovery with limited-subspace weighted matrix factorizations
Modern-day seismic imaging and monitoring technology increasingly rely on
dense full-azimuth sampling. Unfortunately, the costs of acquiring densely
sampled data rapidly become prohibitive and we need to look for ways to
sparsely collect data, e.g. from sparsely distributed ocean bottom nodes, from
which we then derive densely sampled surveys through the method of wavefield
reconstruction. Because of their relatively cheap and simple calculations,
wavefield reconstruction via matrix factorizations has proven to be a viable
and scalable alternative to the more generally used transform-based methods.
While this method is capable of processing all full azimuth data frequency by
frequency slice, its performance degrades at higher frequencies because
monochromatic data at these frequencies is not as well approximated by low-rank
factorizations. We address this problem by proposing a recursive recovery
technique, which involves weighted matrix factorizations where recovered
wavefields at the lower frequencies serve as prior information for the recovery
of the higher frequencies. To limit the adverse effects of potential
overfitting, we propose a limited-subspace recursively weighted matrix
factorization approach where the size of the row and column subspaces to
construct the weight matrices is constrained. We apply our method to data
collected from the Gulf of Suez, and our results show that our limited-subspace
weighted recovery method significantly improves the recovery quality
Transfer learning in large-scale ocean bottom seismic wavefield reconstruction
Achieving desirable receiver sampling in ocean bottom acquisition is often
not possible because of cost considerations. Assuming adequate source sampling
is available, which is achievable by virtue of reciprocity and the use of
modern randomized (simultaneous-source) marine acquisition technology, we are
in a position to train convolutional neural networks (CNNs) to bring the
receiver sampling to the same spatial grid as the dense source sampling. To
accomplish this task, we form training pairs consisting of densely sampled data
and artificially subsampled data using a reciprocity argument and the
assumption that the source-site sampling is dense. While this approach has
successfully been used on the recovery monochromatic frequency slices, its
application in practice calls for wavefield reconstruction of time-domain data.
Despite having the option to parallelize, the overall costs of this approach
can become prohibitive if we decide to carry out the training and recovery
independently for each frequency. Because different frequency slices share
information, we propose the use the method of transfer training to make our
approach computationally more efficient by warm starting the training with CNN
weights obtained from a neighboring frequency slices. If the two neighboring
frequency slices share information, we would expect the training to improve and
converge faster. Our aim is to prove this principle by carrying a series of
carefully selected experiments on a relatively large-scale five-dimensional
data synthetic data volume associated with wide-azimuth 3D ocean bottom node
acquisition. From these experiments, we observe that by transfer training we
are able t significantly speedup in the training, specially at relatively
higher frequencies where consecutive frequency slices are more correlated
Multi-weight Nuclear Norm Minimization for Low-rank Matrix Recovery in Presence of Subspace Prior Information
Weighted nuclear norm minimization has been recently recognized as a
technique for reconstruction of a low-rank matrix from compressively sampled
measurements when some prior information about the column and row subspaces of
the matrix is available. In this work, we study the recovery conditions and the
associated recovery guarantees of weighted nuclear norm minimization when
multiple weights are allowed. This setup might be used when one has access to
prior subspaces forming multiple angles with the column and row subspaces of
the ground-truth matrix. While existing works in this field use a single weight
to penalize all the angles, we propose a multi-weight problem which is designed
to penalize each angle independently using a distinct weight. Specifically, we
prove that our proposed multi-weight problem is stable and robust under weaker
conditions for the measurement operator than the analogous conditions for
single-weight scenario and standard nuclear norm minimization. Moreover, it
provides better reconstruction error than the state of the art methods. We
illustrate our results with extensive numerical experiments that demonstrate
the advantages of allowing multiple weights in the recovery procedure
A Greedy Algorithm for Matrix Recovery with Subspace Prior Information
Matrix recovery is the problem of recovering a low-rank matrix from a few
linear measurements. Recently, this problem has gained a lot of attention as it
is employed in many applications such as Netflix prize problem, seismic data
interpolation and collaborative filtering. In these applications, one might
access to additional prior information about the column and row spaces of the
matrix. These extra information can potentially enhance the matrix recovery
performance. In this paper, we propose an efficient greedy algorithm that
exploits prior information in the recovery procedure. The performance of the
proposed algorithm is measured in terms of the rank restricted isometry
property (R-RIP). Our proposed algorithm with prior subspace information
converges under a more milder condition on the R-RIP in compared with the case
that we do not use prior information. Additionally, our algorithm performs much
better than nuclear norm minimization in terms of both computational complexity
and success rate
Limitations of Implicit Bias in Matrix Sensing: Initialization Rank Matters
In matrix sensing, we first numerically identify the sensitivity to the
initialization rank as a new limitation of the implicit bias of gradient flow.
We will partially quantify this phenomenon mathematically, where we establish
that the gradient flow of the empirical risk is implicitly biased towards
low-rank outcomes and successfully learns the planted low-rank matrix, provided
that the initialization is low-rank and within a specific "capture
neighborhood". This capture neighborhood is far larger than the corresponding
neighborhood in local refinement results; the former contains all models with
zero training error whereas the latter is a small neighborhood of a model with
zero test error. These new insights enable us to design an alternative
algorithm for matrix sensing that complements the high-rank and near-zero
initialization scheme which is predominant in the existing literature
Optimal Weighted Low-rank Matrix Recovery with Subspace Prior Information
Matrix sensing is the problem of reconstructing a low-rank matrix from a few
linear measurements. In many applications such as collaborative filtering, the
famous Netflix prize problem, and seismic data interpolation, there exists some
prior information about the column and row spaces of the ground-truth low-rank
matrix. In this paper, we exploit this prior information by proposing a
weighted optimization problem where its objective function promotes both rank
and prior subspace information. Using the recent results in conic integral
geometry, we obtain the unique optimal weights that minimize the required
number of measurements. As simulation results confirm, the proposed convex
program with optimal weights requires substantially fewer measurements than the
regular nuclear norm minimization
Matrix Completion with Prior Subspace Information via Maximizing Correlation
This paper studies the problem of completing a low-rank matrix from a few of
its random entries with the aid of prior information. We suggest a strategy to
incorporate prior information into the standard matrix completion procedure by
maximizing the correlation between the original signal and the prior
information. We also establish performance guarantees for the proposed method,
which show that with suitable prior information, the proposed procedure can
reduce the sample complexity of the standard matrix completion by a logarithmic
factor. To illustrate the theory, we further analyze an important practical
application where the prior subspace information is available. Both synthetic
and real-world experiments are provided to verify the validity of the theory.Comment: 13 pages, 6 figure
Projection-based QLP Algorithm for Efficiently Computing Low-Rank Approximation of Matrices
Matrices with low numerical rank are omnipresent in many signal processing
and data analysis applications. The pivoted QLP (p-QLP) algorithm constructs a
highly accurate approximation to an input low-rank matrix. However, it is
computationally prohibitive for large matrices. In this paper, we introduce a
new algorithm termed Projection-based Partial QLP (PbP-QLP) that efficiently
approximates the p-QLP with high accuracy. Fundamental in our work is the
exploitation of randomization and in contrast to the p-QLP, PbP-QLP does not
use the pivoting strategy. As such, PbP-QLP can harness modern computer
architectures, even better than competing randomized algorithms. The efficiency
and effectiveness of our proposed PbP-QLP algorithm are investigated through
various classes of synthetic and real-world data matrices
Nonconvex Matrix Completion with Linearly Parameterized Factors
Techniques of matrix completion aim to impute a large portion of missing
entries in a data matrix through a small portion of observed ones, with broad
machine learning applications including collaborative filtering, pairwise
ranking, etc. In practice, additional structures are usually employed in order
to improve the accuracy of matrix completion. Examples include subspace
constraints formed by side information in collaborative filtering, and skew
symmetry in pairwise ranking. This paper performs a unified analysis of
nonconvex matrix completion with linearly parameterized factorization, which
covers the aforementioned examples as special cases. Importantly, uniform upper
bounds for estimation errors are established for all local minima, provided
that the sampling rate satisfies certain conditions determined by the rank,
condition number, and incoherence parameter of the ground-truth low rank
matrix. Empirical efficiency of the proposed method is further illustrated by
numerical simulations
Frank-Wolfe Methods with an Unbounded Feasible Region and Applications to Structured Learning
The Frank-Wolfe (FW) method is a popular algorithm for solving large-scale
convex optimization problems appearing in structured statistical learning.
However, the traditional Frank-Wolfe method can only be applied when the
feasible region is bounded, which limits its applicability in practice.
Motivated by two applications in statistical learning, the trend
filtering problem and matrix optimization problems with generalized nuclear
norm constraints, we study a family of convex optimization problems where the
unbounded feasible region is the direct sum of an unbounded linear subspace and
a bounded constraint set. We propose two new Frank-Wolfe methods: unbounded
Frank-Wolfe method (uFW) and unbounded Away-Step Frank-Wolfe method (uAFW), for
solving a family of convex optimization problems with this class of unbounded
feasible regions. We show that under proper regularity conditions, the
unbounded Frank-Wolfe method has a sublinear convergence rate, and
unbounded Away-Step Frank-Wolfe method has a linear convergence rate, matching
the best-known results for the Frank-Wolfe method when the feasible region is
bounded. Furthermore, computational experiments indicate that our proposed
methods appear to outperform alternative solvers.Comment: 31 pages, 6 figure