BiCoGAN improves cGANs by disentangling latent and auxiliary variables.
problem Improving disentanglement of latent and auxiliary variables in cGANs.
method BiCoGAN uses bidirectional training with extrinsic factor loss and dynamically-tuned importance weight.
result BiCoGAN encodes auxiliary variables more accurately and disentangles latent and auxiliary variables effectively.
Unified tensor model disentangles object appearance factors.
problem Representing hierarchical intrinsic and extrinsic causal factors of object appearance.
method Compositional hierarchical tensor factorization.
result Interpretable object representation robust to occlusion and reduced training data requirements.
Proves conditions for conformal immersions of Riemannian products in Euclidean spaces.
problem Conditions for conformal immersions of Riemannian products in Euclidean spaces.
method Analyzes conditions for conformal immersions of Riemannian products in Euclidean spaces.
result Proves conditions for conformal immersions of Riemannian products in Euclidean spaces.
The paper tackles overfitting in deep neural networks and proposes a consensus-based algorithm to avoid it.
problem Overfitting in deep neural networks.
method Separating correctly and incorrectly classified samples, analyzing dynamics during training, and proposing a consensus-based classification algorithm.
result The consensus-based algorithm significantly improves classification accuracy, especially with limited training samples.
Two methods estimate variable importance; they agree under certain conditions.
problem Quantifying variable importance in machine learning models.
method VIMP and MPLOCO methods for estimating intrinsic and extrinsic variable importance.
result MPLOCO is asymptotically equivalent to VIMP under squared-error loss.
Let $(V, \Om)$ be a symplectic vector space and let $φ: M \ra V$ be a symplectic immersion. We show that φ(M)⊂V is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if and only if the second fundamental form of φ is parallel. Furthermore, we show that any symmetric spa…
New bounds on knot distortion and Seifert surface properties.
problem Understanding the distortion of knots and properties of Seifert surfaces.
method Analyzing embeddings of Seifert surfaces and using properties of monodromy maps.
result Bounds on the distortion of certain knots and properties of Seifert surfaces.
Study shows startup competition and investor network influence fundraising success at different stages.
problem Misunderstanding of fundraising success factors across startup stages.
method Used Word2Vec for competition measures and Graph Neural Networks for investor network analysis.
result Startup competition is crucial for early-stage fundraising, while growth-stage fundraising is influenced by investor network features.
Uniqueness of 1D bi-Schrödinger flow proven from flat torus to compact space.
problem Proving uniqueness of a smooth flow from flat torus to compact space.
method Extrinsic approach using isometric embedding into Euclidean space, modifying classical H2-energy to handle loss of derivatives. result Uniqueness of the generalized bi-Schrödinger flow established.
Derives GJMS operators and Q-curvatures for submanifolds.
problem Understanding geometric properties of submanifolds in conformal manifolds.
method Realizes conformal manifold as Poincaré-Einstein space boundary, derives operators as obstructions, uses ambient metric for conformal invariance.
result Explicit formulas and factorization for GJMS operators of orders 2 and 4, conformal invariance for all orders in all dimensions.
Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.
problem Analyzing Q-curvatures and Paneitz operators for hypersurfaces.
method Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.
result Explicit formulas for the extrinsic Paneitz operators P_4 and extrinsic Q-curvatures for totally umbilic hypersurfaces in any dimension.
The study defines and characterizes extrinsic catenaries in hyperbolic space.
problem Understanding catenaries in hyperbolic geometry.
method Defined extrinsic catenaries in hyperbolic plane, characterized them, and proved their relation to minimal surfaces.
result Extrinsic catenaries in hyperbolic space are critical points of a potential functional and generating curves of minimal surfaces.
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.
Hedonic models predict 84-92% of U.S. real estate prices, highlighting environmental factors' impact.
problem Predicting real estate prices using hedonic models with environmental factors.
method P-spline generalized additive models for real estate prices, contrasting with linear and polynomial models.
result GAM models explain 84-92% of U.S. real estate price variance, with environmental factors contributing minimally.
We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding …
DSM on manifolds removes singularities and computes small-noise expansions.
problem DSM on manifolds with singular noise.
method Rao-Blackwellized score matching, nearest-point projection, intrinsic Riemannian score.
result Canonical target equals intrinsic Riemannian score up to a small correction.
Flat tori in 3-sphere have π extrinsic diameter under specific conditions.
problem Determining the extrinsic diameter of immersed flat tori in the 3-sphere.
method Analyzing asymptotic curves and Hopf fibration projections.
result The extrinsic diameter is π under certain topological conditions.
Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…
Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.
problem Understanding the relationship between intrinsic and extrinsic metrics and surface area.
method Examined surfaces within a unit ball in R3, provided lower bounds on the ratio in terms of area, and showed non-existence of global lower bounds.
result Found that the ratio of intrinsic to extrinsic metrics has a lower bound in terms of surface area, but no global lower bound exists.
Bounds on Dirac operators linked to manifold properties.
problem Estimating Dirac operators on submanifolds.
method Lower and upper bounds using conformal and extrinsic quantities.
result Proved bounds for Dirac operators.
New operators and curvatures derived from embedded manifolds.
problem Finding obstructions and coupling extrinsic operators.
method Explicit computation of extrinsic Paneitz operator and its applications.
result New extrinsically-coupled fourth and sixth order operators.
Agent learns independently controllable factors by interacting with its environment.
problem Discovering independent controllable factors in environments.
method Proposes a training framework for an agent to interact with its environment and learn independently controllable factors.
result Agent can disentangle independently controllable aspects of the environment without extrinsic reward.
Analyzes canonical reductive decomposition of extrinsic homogeneous submanifolds.
problem Understanding the reductive decomposition of extrinsic homogeneous submanifolds.
method Examines Lie subgroups and reductive decompositions of homogeneous structures.
result Establishes a connection with the Ambrose-Singer theorem and homogeneous structures.
In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures.
Novel coarse extrinsic curvature for Riemannian submanifolds.
problem Understanding extrinsic curvature of submanifolds.
method Derived from Wasserstein 1-distance between probability measures.
result New insights and approximation of mean curvature from data.
The paper studies the loss landscape of regularized deep matrix factorization, revealing unique and sharp minimizers.
problem Understanding the loss landscape and minimizers of regularized deep matrix factorization problems.
method Theoretical analysis of ℓ2-regularized deep matrix factorization/deep linear network training problems with squared-error loss. result The unique end-to-end minimizer exists for all target matrices except for a set of Lebesgue measure zero.
We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.
Agent learns independently controllable factors by interacting with environment.
problem Discovering independently controllable factors of variation in interactions with the world.
method Proposed a training framework for an agent to interact with the environment, hypothesizing that independently controllable factors correspond to aspects of the environment that can be manipulated.
result Agent can disentangle independently controllable aspects of the environment without extrinsic reward signals.
We study the topology of (properly) immersed complete minimal surfaces P2 in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…
Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…
We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.
In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.
The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…
Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…
The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N o…
Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic st…
Derives formulas for extrinsic Paneitz operator and Q-curvature in general dimensions.
problem Calculating extrinsic conformal invariants for hypersurfaces in Riemannian manifolds.
method Explicit formulas derived using local conformal invariants and non-trivial local conformal invariant C. result Explicit formulas for extrinsic Paneitz operator and Q-curvature in general dimensions. We obtain an optimal estimate for the extrinsic curvature of an entire minimal graph in $\H^2\times\R$, $\H^2$ the hyperbolic plane.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature κ≤0. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the subm…
Paper proves a Penrose inequality in extrinsic geometry.
problem Proving a Penrose inequality in extrinsic geometry.
method Analyzing minimal capillary surfaces and their free energy.
result Established an extrinsic Penrose inequality.
Paper finds convexity in translating solitons for concave flows.
problem Understanding convexity in translating solitons for concave extrinsic flows.
method Analyzes convexity estimates for translating solitons evolving under concave functions in Rn+1. result Establishes convexity estimates for translating solitons of concave extrinsic geometric flows.
Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.
problem Proving anisotropic extrinsic radius pinching inequality for hypersurfaces.
method Analyzes anisotropic mean curvatures and studies equality cases.
result Almost extremal hypersurfaces are close to Wulff shape.
Method estimates exogenous and endogenous factors from event times.
problem Estimating factors influencing event occurrence.
method Combines inhomogeneous Poisson and Hawkes processes, fits using free energy minimization.
result Four regimes identified based on factor detection.
Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.
problem Classifying quasicomplete surfaces in 3-space-forms.
method Using quasicompleteness as a weaker form of completeness, the global geometry of surfaces is determined.
result Geodesic spheres are the only quasicomplete surfaces of constant extrinsic curvature in 3-space-forms.
Researchers define residue families and use them to solve singular Yamabe problems.
problem Solving singular Yamabe problems on manifolds with boundary.
method Introducing residue families and using them to construct differential operators.
result Residue families can be written as compositions of degenerate Laplacians for approximate solutions of singular Yamabe problems.