114 research outputs found
Statistical and computational rates in high rank tensor estimation
Higher-order tensor datasets arise commonly in recommendation systems,
neuroimaging, and social networks. Here we develop probable methods for
estimating a possibly high rank signal tensor from noisy observations. We
consider a generative latent variable tensor model that incorporates both high
rank and low rank models, including but not limited to, simple hypergraphon
models, single index models, low-rank CP models, and low-rank Tucker models.
Comprehensive results are developed on both the statistical and computational
limits for the signal tensor estimation. We find that high-dimensional latent
variable tensors are of log-rank; the fact explains the pervasiveness of
low-rank tensors in applications. Furthermore, we propose a polynomial-time
spectral algorithm that achieves the computationally optimal rate. We show that
the statistical-computational gap emerges only for latent variable tensors of
order 3 or higher. Numerical experiments and two real data applications are
presented to demonstrate the practical merits of our methods.Comment: 38 pages, 8 figure
The Structuralist Approach to Underdetermination
This paper provides an exposition of the structuralist approach to underdetermination, which aims to resolve the underdetermination of theories by identifying their common theoretical structure. Applications of the structuralist approach can be found in many areas of philosophy. I present a schema of the structuralist approach, which conceptually unifies such applications in different subject matters. It is argued that two classic arguments in the literature, Paul Benacerraf’s argument on natural numbers and W. V. O. Quine’s argument for the indeterminacy of translation, can be analyzed as instances of the structuralist schema. These two applications illustrate different kinds of conclusions that can be drawn through the structuralist approach; Benacerraf’s argument shows that we can derive an ontological conclusion about the given subject matter, while Quine’s structuralist approach leads to a semantic conclusion about how to determine linguistic meanings given radical translation. Then, as a case study, I review a recent debate in metaphysics between Shamik Dasgupta, Jason Turner, and Catharine Diehl to consider the extent to which different instances of the structuralist schema are conceptually unified. Both sides of the debate can be interpreted as utilizing the structuralist approach; one side uses the structuralist approach for an ontological conclusion, while the other side relies on a semantic conclusion. I argue that this has a strong dialectical consequence, which sheds light on the conceptual unity of the structuralist approach
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What Analytic Metaphysics Can Do For Scientific Metaphysics
The apparent chasm between two camps in metaphysics, analytic metaphysics and scientific metaphysics, is well recognized. I argue that the relationship between them is not necessarily a rivalry; a division of labour that resembles the relationship between pure mathematics and science is possible. As a case study, I look into the metaphysical underdetermination argument for ontic structural realism, a well-known position in scientific metaphysics, together with an argument for the position in analytic metaphysics known as ontological nihilism. I argue that we can ascribe the same schema to both arguments, which indicates that analytic metaphysics can offer an abstract model that scientific metaphysics may find useful
Dynamical Billiard and a long-time behavior of the Boltzmann equation in general 3D toroidal domains
Establishing global well-posedness and convergence toward equilibrium of the
Boltzmann equation with specular reflection boundary condition has been one of
the central questions in the subject of kinetic theory. Despite recent
significant progress in this question when domains are strictly convex, as
shown by Guo and Kim-Lee, the same question without the strict convexity of
domains is still totally open in 3D. The major difficulty arises when a
billiard map has an infinite number of bounces in a finite time interval or
when the map fails to be Lipschitz continuous, both of which happen generically
when the domain is non-convex. In this paper, we develop a new method to
control a billiard map on a surface of revolution generated by revolving any
planar analytic convex closed curve (e.g., typical shape of tokamak reactors'
chamber). In particular, we classify and measure the size (to be small) of a
pullback set (along the billiard trajectory) of the infinite-bouncing and
singular-bouncing cases. As a consequence, we solve the open question
affirmatively in such domains. To the best of our knowledge, this work is the
first construction of global solutions to the hard-sphere Boltzmann equation in
generic non-convex 3-dimensional domains.Comment: 97 pages, 11 figure
The structuralist approach to underdetermination
This paper provides an exposition of the structuralist approach to underdetermination, which aims to resolve the underdetermination of theories by identifying their common theoretical structure. Applications of the structuralist approach can be found in many areas of philosophy. I present a schema of the structuralist approach, which conceptually unifies such applications in different subject matters. It is argued that two classic arguments in the literature, Paul Benacerraf’s argument on natural numbers and W. V. O. Quine’s argument for the indeterminacy of translation, can be analyzed as instances of the structuralist schema. These two applications illustrate different kinds of conclusions that can be drawn through the structuralist approach; Benacerraf’s argument shows that we can derive an ontological conclusion about the given subject matter, while Quine’s structuralist approach leads to a semantic conclusion about how to determine linguistic meanings given radical translation. Then, as a case study, I review a recent debate in metaphysics between Shamik Dasgupta, Jason Turner, and Catharine Diehl to consider the extent to which different instances of the structuralist schema are conceptually unified. Both sides of the debate can be interpreted as utilizing the structuralist approach; one side uses the structuralist approach for an ontological conclusion, while the other side relies on a semantic conclusion. I argue that this has a strong dialectical consequence, which sheds light on the conceptual unity of the structuralist approach
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