11 research outputs found
Manifold Constrained Low-Rank Decomposition
Low-rank decomposition (LRD) is a state-of-the-art method for visual data
reconstruction and modelling. However, it is a very challenging problem when
the image data contains significant occlusion, noise, illumination variation,
and misalignment from rotation or viewpoint changes. We leverage the specific
structure of data in order to improve the performance of LRD when the data are
not ideal. To this end, we propose a new framework that embeds manifold priors
into LRD. To implement the framework, we design an alternating direction method
of multipliers (ADMM) method which efficiently integrates the manifold
constraints during the optimization process. The proposed approach is
successfully used to calculate low-rank models from face images, hand-written
digits and planar surface images. The results show a consistent increase of
performance when compared to the state-of-the-art over a wide range of
realistic image misalignments and corruptions
Robust Principal Component Analysis on Graphs
Principal Component Analysis (PCA) is the most widely used tool for linear
dimensionality reduction and clustering. Still it is highly sensitive to
outliers and does not scale well with respect to the number of data samples.
Robust PCA solves the first issue with a sparse penalty term. The second issue
can be handled with the matrix factorization model, which is however
non-convex. Besides, PCA based clustering can also be enhanced by using a graph
of data similarity. In this article, we introduce a new model called "Robust
PCA on Graphs" which incorporates spectral graph regularization into the Robust
PCA framework. Our proposed model benefits from 1) the robustness of principal
components to occlusions and missing values, 2) enhanced low-rank recovery, 3)
improved clustering property due to the graph smoothness assumption on the
low-rank matrix, and 4) convexity of the resulting optimization problem.
Extensive experiments on 8 benchmark, 3 video and 2 artificial datasets with
corruptions clearly reveal that our model outperforms 10 other state-of-the-art
models in its clustering and low-rank recovery tasks
Low-Rank Matrices on Graphs: Generalized Recovery & Applications
Many real world datasets subsume a linear or non-linear low-rank structure in
a very low-dimensional space. Unfortunately, one often has very little or no
information about the geometry of the space, resulting in a highly
under-determined recovery problem. Under certain circumstances,
state-of-the-art algorithms provide an exact recovery for linear low-rank
structures but at the expense of highly inscalable algorithms which use nuclear
norm. However, the case of non-linear structures remains unresolved. We revisit
the problem of low-rank recovery from a totally different perspective,
involving graphs which encode pairwise similarity between the data samples and
features. Surprisingly, our analysis confirms that it is possible to recover
many approximate linear and non-linear low-rank structures with recovery
guarantees with a set of highly scalable and efficient algorithms. We call such
data matrices as \textit{Low-Rank matrices on graphs} and show that many real
world datasets satisfy this assumption approximately due to underlying
stationarity. Our detailed theoretical and experimental analysis unveils the
power of the simple, yet very novel recovery framework \textit{Fast Robust PCA
on Graphs
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Data-driven imputation of miscibility of aqueous solutions via graph-regularized logistic matrix factorization
Aqueous, two-phase systems (ATPSs) may form upon mixing two solutions of independently water-soluble compounds. Many separation, purification, and extraction processes rely on ATPSs. Predicting the miscibility of solutions can accelerate and reduce the cost of the discovery of new ATPSs for these applications. Whereas previous machine learning approaches to ATPS prediction used physicochemical properties of each solute as a descriptor, in this work, we show how to impute missing miscibility outcomes directly from an incomplete collection of pairwise miscibility experiments. We use graph-regularized logistic matrix factorization (GR-LMF) to learn a latent vector of each solution from (i) the observed entries in the pairwise miscibility matrix and (ii) a graph (nodes: solutes, edges: shared relationships) indicating the general category of the solute (i.e., polymer, surfactant, salt, protein). For an experimental dataset of the pairwise miscibility of 68 solutions from Peacock et al. [ACS Appl. Mater. Interfaces 2021, 13, 11449–11460], we find that GR-LMF more accurately predicts missing (im)miscibility outcomes of pairs of solutions than ordinary logistic matrix factorization and random forest classifiers that use physicochemical features of the solutes. GR-LMF obviates the need for features of the solutions/solutes to impute missing miscibility outcomes, but it cannot predict the miscibility of a new solution without some observations of its miscibility with other solutions in the training data set
Curvature corrected tangent space-based approximation of manifold-valued data
When generalizing schemes for real-valued data approximation or decomposition
to data living in Riemannian manifolds, tangent space-based schemes are very
attractive for the simple reason that these spaces are linear. An open
challenge is to do this in such a way that the generalized scheme is applicable
to general Riemannian manifolds, is global-geometry aware and is
computationally feasible. Existing schemes have been unable to account for all
three of these key factors at the same time.
In this work, we take a systematic approach to developing a framework that is
able to account for all three factors. First, we will restrict ourselves to the
-- still general -- class of symmetric Riemannian manifolds and show how
curvature affects general manifold-valued tensor approximation schemes. Next,
we show how the latter observations can be used in a general strategy for
developing approximation schemes that are also global-geometry aware. Finally,
having general applicability and global-geometry awareness taken into account
we restrict ourselves once more in a case study on low-rank approximation. Here
we show how computational feasibility can be achieved and propose the
curvature-corrected truncated higher-order singular value decomposition
(CC-tHOSVD), whose performance is subsequently tested in numerical experiments
with both synthetic and real data living in symmetric Riemannian manifolds with
both positive and negative curvature