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We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on…
This article provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed by W.Schmid in [Sch]. A corresponding problem in the compact group setting was solved by…
Localization properties of scalar single particle states are analyzed by explicit calculational examples with a focus on the massless case. Problems arising from the non-existence of relativistic particle position operators respecting the…
We show that moduli spaces of stable maps admits virtual orbifold structure. The symplectic version of virtual localization formula is obtained.
We discuss the theory of equivariant localization focussing on applications relevant for holography. We consider geometries comprising compact and non-compact toric orbifolds, as well as more general non-compact toric Calabi-Yau…
We consider Killing vector fields on standard static space-times and obtain equations for a vector field on a standard static space-time to be Killing. We also provide a characterization of Killing vector fields on standard static…
In this article we give a totally new proof of the integral localization formula for equivariantly closed differential forms (Theorem 7.11 in [BGV]). We restate it here as Theorem 2. This localization formula is very well known, but the…
We describe the applications of localization methods, in particular the functorial localization formula, in the proofs of several conjectures from string theory. Functorial localization formula pushes the computations on complicated moduli…
In this paper we propose a theory of contact invariants and open string invariants, which are generalizations of the relative invariants. We introduce two moduli spaces $\bar{\mathcal{M}}_{A}(M^{+},C,g,m+\nu,{\bf y},{\bf…
A promising approach to accurate positioning of robots is ground texture based localization. It is based on the observation that visual features of ground images enable fingerprint-like place recognition. We tackle the issue of efficient…
We present the local classification of singularities of smooth vector fields on the line, with respect to the equivalence relation of $C^1$--conjugacy. Along the way, we recall the analogous classification, up to $C^0$ and $C^{\infty}$…
Multidimensional factorization method is formulated in arbitrary curvilinear coordinates. Particular cases of polar and spherical coordinates are considered and matrix potentials with separating variables are constructed. A new class of…
This paper proposes a general machine learning framework called the localization method, which is fundamentally built on two core concepts: localization kernels and local means -- key components that underpin the self-attention mechanism.…
In this work we classify the at-point regularities of set-valued mappings into two categories and then we analyze their relationship through several implications and examples. After this theoretical tour, we use the subregularity properties…
Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…
By double ideal quotient, we mean $(I:(I:J))$ where ideals $I$ and $J$. In our previous work [11], double ideal quotient and its variants are shown to be very useful for checking prime divisor and generating primary component. Combining…
Invariant coordinate selection is an unsupervised multivariate data transformation useful in many contexts such as outlier detection or clustering. It is based on the simultaneous diagonalization of two affine equivariant and positive…
In this article I describe my recent geometric localization argument dealing with actions of NONcompact groups which provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the…
Locally inertial coordinates are constructed by carrying Riemann normal coordinates on a codimension two spacelike surface along the geodesics normal to it. Since the normal tangents are labelled by components with respect to a null basis,…
Global mobile robot localization is the problem of determining a robot's pose in an environment, using sensor data, when the starting position is unknown. A family of probabilistic algorithms known as Monte Carlo Localization (MCL) is…