100 research outputs found
Adaptive Graph via Multiple Kernel Learning for Nonnegative Matrix Factorization
Nonnegative Matrix Factorization (NMF) has been continuously evolving in
several areas like pattern recognition and information retrieval methods. It
factorizes a matrix into a product of 2 low-rank non-negative matrices that
will define parts-based, and linear representation of nonnegative data.
Recently, Graph regularized NMF (GrNMF) is proposed to find a compact
representation,which uncovers the hidden semantics and simultaneously respects
the intrinsic geometric structure. In GNMF, an affinity graph is constructed
from the original data space to encode the geometrical information. In this
paper, we propose a novel idea which engages a Multiple Kernel Learning
approach into refining the graph structure that reflects the factorization of
the matrix and the new data space. The GrNMF is improved by utilizing the graph
refined by the kernel learning, and then a novel kernel learning method is
introduced under the GrNMF framework. Our approach shows encouraging results of
the proposed algorithm in comparison to the state-of-the-art clustering
algorithms like NMF, GrNMF, SVD etc.Comment: This paper has been withdrawn by the author due to the terrible
writin
The GSVD: Where are the ellipses?, Matrix Trigonometry, and more
This paper provides an advanced mathematical theory of the Generalized
Singular Value Decomposition (GSVD) and its applications. We explore the
geometry of the GSVD which provides a long sought for ellipse picture which
includes a horizontal and a vertical multiaxis. We further propose that the
GSVD provides natural coordinates for the Grassmann manifold. This paper proves
a theorem showing how the finite generalized singular values do or do not
relate to the singular values of .
We then turn to the applications arguing that this geometrical theory is
natural for understanding existing applications and recognizing opportunities
for new applications. In particular the generalized singular vectors play a
direct and as natural a mathematical role for certain applications as the
singular vectors do for the SVD. In the same way that experts on the SVD often
prefer not to cast SVD problems as eigenproblems, we propose that the GSVD,
often cast as a generalized eigenproblem, is rather best cast in its natural
setting.
We illustrate this theoretical approach and the natural multiaxes (with
labels from technical domains) in the context of applications where the GSVD
arises: Tikhonov regularization (unregularized vs regularization), Genome
Reconstruction (humans vs yeast), Signal Processing (signal vs noise), and
stastical analysis such as ANOVA and discriminant analysis (between clusters vs
within clusters.) With the aid of our ellipse figure, we encourage in the
future the labelling of the natural multiaxes in any GSVD problem.Comment: 28 page
Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties
The two-dimensional supersymmetric gauged linear sigma model (GLSM) with
abelian gauge groups and matter fields has provided many insights into string
theory on Calabi--Yau manifolds of a certain type: complete intersections in
toric varieties. In this paper, we consider two GLSM constructions with
nonabelian gauge groups and charged matter whose infrared CFTs correspond to
string propagation on determinantal Calabi-Yau varieties, furnishing another
broad class of Calabi-Yau geometries in addition to complete intersections. We
show that these two models -- which we refer to as the PAX and the PAXY model
-- are dual descriptions of the same low-energy physics. Using GLSM techniques,
we determine the quantum K\"ahler moduli space of these varieties and find no
disagreement with existing results in the literature.Comment: v3: 46 pages, 1 figure. Corrected phase structure of general linear
determinantal varieties. Typos correcte
Algebras of commuting differential operators for integral kernels of Airy type
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux transformations of the Airy function in terms of the fixed point submanifold of a differential Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy Adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with each of the integral operators in question. We leverage the general theory of [8] to obtain explicit formulas for the two differential operators of lowest orders that commute with each of the level one and two integral operators obtained in the Darboux process. Moreover, we prove that each pair of differential operators commute with each other. The commuting operators in the level one case are shown to satisfy an algebraic relation defining an elliptic curve.Fil: Casper, W. Riley. California State University, Fullerton; Estados UnidosFil: Grünbaum, Francisco Alberto. University of California at Berkeley; Estados UnidosFil: Yakimov, Milen. Northeastern University; Estados UnidosFil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin
Apprentissage de représentation pour des données générées par des utilisateurs
In this thesis, we study how representation learning methods can be applied to user-generated data. Our contributions cover three different applications but share a common denominator: the extraction of relevant user representations. Our first application is the item recommendation task, where recommender systems build user and item profiles out of past ratings reflecting user preferences and item characteristics. Nowadays, textual information is often together with ratings available and we propose to use it to enrich the profiles extracted from the ratings. Our hope is to extract from the textual content shared opinions and preferences. The models we propose provide another opportunity: predicting the text a user would write on an item. Our second application is sentiment analysis and, in particular, polarity classification. Our idea is that recommender systems can be used for such a task. Recommender systems and traditional polarity classifiers operate on different time scales. We propose two hybridizations of these models: the former has better classification performance, the latter highlights a vocabulary of surprise in the texts of the reviews. The third and final application we consider is urban mobility. It takes place beyond the frontiers of the Internet, in the physical world. Using authentication logs of the subway users, logging the time and station at which users take the subway, we show that it is possible to extract robust temporal profiles.Dans cette thèse, nous étudions comment les méthodes d'apprentissage de représentations peuvent être appliquées à des données générées par l'utilisateur. Nos contributions couvrent trois applications différentes, mais partagent un dénominateur commun: l'extraction des représentations d'utilisateurs concernés. Notre première application est la tâche de recommandation de produits, où les systèmes existant créent des profils utilisateurs et objets qui reflètent les préférences des premiers et les caractéristiques des derniers, en utilisant l'historique. De nos jours, un texte accompagne souvent cette note et nous proposons de l'utiliser pour enrichir les profils extraits. Notre espoir est d'en extraire une connaissance plus fine des goûts des utilisateurs. Nous pouvons, en utilisant ces modèles, prédire le texte qu'un utilisateur va écrire sur un objet. Notre deuxième application est l'analyse des sentiments et, en particulier, la classification de polarité. Notre idée est que les systèmes de recommandation peuvent être utilisés pour une telle tâche. Les systèmes de recommandation et classificateurs de polarité traditionnels fonctionnent sur différentes échelles de temps. Nous proposons deux hybridations de ces modèles: la première a de meilleures performances en classification, la seconde exhibe un vocabulaire de surprise. La troisième et dernière application que nous considérons est la mobilité urbaine. Elle a lieu au-delà des frontières d'Internet, dans le monde physique. Nous utilisons les journaux d'authentification des usagers du métro, enregistrant l'heure et la station d'origine des trajets, pour caractériser les utilisateurs par ses usages et habitudes temporelles
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