## Seminar information archive

Seminar information archive ～02/22｜Today's seminar 02/23 | Future seminars 02/24～

### 2016/01/08

#### Colloquium

16:50-17:50 Room #123 (Graduate School of Math. Sci. Bldg.)

Birational geometry through complex dymanics (ENGLISH)

**Keiji Oguiso**(Graduate School of Mathematical Sciences, University of Tokyo)Birational geometry through complex dymanics (ENGLISH)

[ Abstract ]

Birational geometry and complex dymanics are rich subjects having

interactions with many branches of mathematics. On the other hand,

though these two subjects share many common interests hidden especially

when one considers group symmetry of manifolds, it seems rather recent

that their rich interations are really notified, perhaps after breaking

through works for surface automorphisms in the view of topological

entropy by Cantat and McMullen early in this century, by which I was so

mpressed.

The notion of entropy of automorphism is a fundamental invariant which

measures how fast two general points spread out fast under iteration. So,

the exisitence of surface automorphism of positive entropy with Siegel

disk due to McMullen was quite surprizing. The entropy also measures, by

the fundamenal theorem of Gromov-Yomdin, the

logarithmic growth of the degree of polarization under iteration. For

instance, the Mordell-Weil group of an elliptic fibration is a very

intersting rich subject in algebraic geometry and number theory, but the

group preserves the fibration so that it might not be so interesting

from dynamical view point. However, if the surface admits two different

elliptic fibrations, which often happens in K3 surfaces of higher Picard

number, then highly non-commutative dynamically rich phenomena can be

observed.

In this talk, I would like to explain the above mentioned phenomena with

a few unexpected applications that I noticed in these years:

(1) Kodaira problem on small deformation of compact Kaehler manifolds in

higher dimension via K3 surface automorphism with Siegel disk;

(2) Geometric liftability problem of automorphisms in positive

characteristic to chacateristic 0 via Mordell-Weil groups and number

theoretic aspect of entropy;

(3) Existence problem on primitive automorphisms of projective manifolds,

through (relative) dynamical degrees due to Dinh-Sibony, Dinh-Nguyen-

Troung, a powerful refinement of the notion of entropy, with by-product

for Ueno-Campana's problem on (uni)rationality of manifolds of torus

quotient.

Birational geometry and complex dymanics are rich subjects having

interactions with many branches of mathematics. On the other hand,

though these two subjects share many common interests hidden especially

when one considers group symmetry of manifolds, it seems rather recent

that their rich interations are really notified, perhaps after breaking

through works for surface automorphisms in the view of topological

entropy by Cantat and McMullen early in this century, by which I was so

mpressed.

The notion of entropy of automorphism is a fundamental invariant which

measures how fast two general points spread out fast under iteration. So,

the exisitence of surface automorphism of positive entropy with Siegel

disk due to McMullen was quite surprizing. The entropy also measures, by

the fundamenal theorem of Gromov-Yomdin, the

logarithmic growth of the degree of polarization under iteration. For

instance, the Mordell-Weil group of an elliptic fibration is a very

intersting rich subject in algebraic geometry and number theory, but the

group preserves the fibration so that it might not be so interesting

from dynamical view point. However, if the surface admits two different

elliptic fibrations, which often happens in K3 surfaces of higher Picard

number, then highly non-commutative dynamically rich phenomena can be

observed.

In this talk, I would like to explain the above mentioned phenomena with

a few unexpected applications that I noticed in these years:

(1) Kodaira problem on small deformation of compact Kaehler manifolds in

higher dimension via K3 surface automorphism with Siegel disk;

(2) Geometric liftability problem of automorphisms in positive

characteristic to chacateristic 0 via Mordell-Weil groups and number

theoretic aspect of entropy;

(3) Existence problem on primitive automorphisms of projective manifolds,

through (relative) dynamical degrees due to Dinh-Sibony, Dinh-Nguyen-

Troung, a powerful refinement of the notion of entropy, with by-product

for Ueno-Campana's problem on (uni)rationality of manifolds of torus

quotient.

### 2016/01/06

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Quantum channels from the free orthogonal quantum group (English)

**Benoit Collins**(Kyoto Univ.)Quantum channels from the free orthogonal quantum group (English)

### 2016/01/05

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Stationary scattering theory on manifolds (English)

**Eric Skibsted**(Aarhus University, Denmark)Stationary scattering theory on manifolds (English)

[ Abstract ]

We present a stationary scattering theory for the Schrödinger operator on Riemannian manifolds with the structure of ends each of which is equipped with an escape function (for example a convex distance function). This includes manifolds with ends modeled as cone-like subsets of the Euclidean space and/or the hyperbolic space. Our results include Rellich’s theorem, the limiting absorption principle, radiation condition bounds, the Sommerfeld uniqueness result, and we give complete characterization/asymptotics of the generalized eigenfunctions in a certain Besov space and show asymptotic completeness (with K. Ito).

We present a stationary scattering theory for the Schrödinger operator on Riemannian manifolds with the structure of ends each of which is equipped with an escape function (for example a convex distance function). This includes manifolds with ends modeled as cone-like subsets of the Euclidean space and/or the hyperbolic space. Our results include Rellich’s theorem, the limiting absorption principle, radiation condition bounds, the Sommerfeld uniqueness result, and we give complete characterization/asymptotics of the generalized eigenfunctions in a certain Besov space and show asymptotic completeness (with K. Ito).

### 2015/12/21

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Scaling limits of random walks on trees (English)

**David Croydon**(University of Warwick)Scaling limits of random walks on trees (English)

[ Abstract ]

I will survey some recent work regarding the scaling limits of random walks on trees, as well as the scaling of the associated local times and cover time. The trees considered will include self-similar pre-fractal graphs, critical Galton-Watson trees and the uniform spanning tree in two dimensions.

I will survey some recent work regarding the scaling limits of random walks on trees, as well as the scaling of the associated local times and cover time. The trees considered will include self-similar pre-fractal graphs, critical Galton-Watson trees and the uniform spanning tree in two dimensions.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On pseudo Kobayashi hyperbolicity of subvarieties of abelian varieties

(Japanese)

**Katsutoshi Yamanoi**(Osaka Univ.)On pseudo Kobayashi hyperbolicity of subvarieties of abelian varieties

(Japanese)

[ Abstract ]

A subvariety of an abelian variety is of general type if and only if it is pseudo Kobayashi hyperbolic. I will discuss the proof of this result.

A subvariety of an abelian variety is of general type if and only if it is pseudo Kobayashi hyperbolic. I will discuss the proof of this result.

### 2015/12/17

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Polarization and stability on a derived equivalent abelian variety (English)

http://db.ipmu.jp/member/personal/3989en.html

**Dulip Piyaratne**(IPMU)Polarization and stability on a derived equivalent abelian variety (English)

[ Abstract ]

In this talk I will explain how one can define a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. Furthermore, we see how such a realisations is connected with stability conditions on their derived categories. Then I will discuss these ideas for abelian surfaces and abelian 3-folds in detail.

[ Reference URL ]In this talk I will explain how one can define a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. Furthermore, we see how such a realisations is connected with stability conditions on their derived categories. Then I will discuss these ideas for abelian surfaces and abelian 3-folds in detail.

http://db.ipmu.jp/member/personal/3989en.html

### 2015/12/16

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Nuclearity in AQFT and related results

**Yul Otani**(Univ. Tokyo)Nuclearity in AQFT and related results

#### thesis presentations

10:30-11:45 Room #122 (Graduate School of Math. Sci. Bldg.)

Special Lagrangian submanifolds and mean curvature flows（特殊ラグランジュ部分多様体と平均曲率流について） (JAPANESE)

**山本 光**(東京大学大学院数理科学研究科)Special Lagrangian submanifolds and mean curvature flows（特殊ラグランジュ部分多様体と平均曲率流について） (JAPANESE)

#### FMSP Lectures

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Selected topics in fractional partial differential equations (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Luchko.pdf

**Yuri Luchko**(University of Applied Sciences, Berlin)Selected topics in fractional partial differential equations (ENGLISH)

[ Abstract ]

In this talk, some remarkable mathematical and physical properties of solutions to the fractional diffusion equation, the alpha-fractional diffusion and alpha-fractional wave equations, the fractional reaction-diffusion equation, and the fractional Schrödinger equation are revisited. From the mathematical viewpoint, the maximum principle for the initial-boundary-value problems for the fractional diffusion equation, the scaling properties of the solutions to the alpha-fractional diffusion and alpha-fractional wave equations and the role of the Mellin integral transform technique for their analytical treatment, as well as the eigenvalue problem for the fractional Schrödinger equation are considered. Physical aspects include a discussion of a probabilistic interpretation of the fundamental solutions to the Cauchy problem for the alpha-fractional diffusion equation, their entropy and the entropy production rates, and some different concepts of the propagation velocities of the fractional wave processes.

[ Reference URL ]In this talk, some remarkable mathematical and physical properties of solutions to the fractional diffusion equation, the alpha-fractional diffusion and alpha-fractional wave equations, the fractional reaction-diffusion equation, and the fractional Schrödinger equation are revisited. From the mathematical viewpoint, the maximum principle for the initial-boundary-value problems for the fractional diffusion equation, the scaling properties of the solutions to the alpha-fractional diffusion and alpha-fractional wave equations and the role of the Mellin integral transform technique for their analytical treatment, as well as the eigenvalue problem for the fractional Schrödinger equation are considered. Physical aspects include a discussion of a probabilistic interpretation of the fundamental solutions to the Cauchy problem for the alpha-fractional diffusion equation, their entropy and the entropy production rates, and some different concepts of the propagation velocities of the fractional wave processes.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Luchko.pdf

### 2015/12/15

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The Curved Cartan Complex (ENGLISH)

**Constantin Teleman**(University of California, Berkeley)The Curved Cartan Complex (ENGLISH)

[ Abstract ]

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant polynomial

functions on the Lie algebra with values in differential forms and a deformation

of the de Rham differential. Before extracting invariants, the Cartan differential

does not square to zero and is apparently meaningless. Unrecognised was the fact

that the full complex is a curved algebra, computing the quotient by G of the

algebra of differential forms on X. This generates, for example, a gauged version of

string topology. Another instance of the construction, applied to deformation

quantisation of symplectic manifolds, gives the BRST construction of the symplectic

quotient. Finally, the theory for a X point with an additional quadratic curving

computes the representation category of the compact group G, and this generalises

to the loop group of G and even to real semi-simple groups.

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant polynomial

functions on the Lie algebra with values in differential forms and a deformation

of the de Rham differential. Before extracting invariants, the Cartan differential

does not square to zero and is apparently meaningless. Unrecognised was the fact

that the full complex is a curved algebra, computing the quotient by G of the

algebra of differential forms on X. This generates, for example, a gauged version of

string topology. Another instance of the construction, applied to deformation

quantisation of symplectic manifolds, gives the BRST construction of the symplectic

quotient. Finally, the theory for a X point with an additional quadratic curving

computes the representation category of the compact group G, and this generalises

to the loop group of G and even to real semi-simple groups.

### 2015/12/14

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Twistor correspondence for associative Grassmanniann

**Fuminori Nakata**(Fukushima Univ.)Twistor correspondence for associative Grassmanniann

[ Abstract ]

It is well known that the 6-dimensional sphere has a non-integrable almost complex structure which is introduced from the (right) multiplication of imaginary octonians. On this 6-sphere, there is a family of psuedo-holomorphic $\mathbb{C}\mathbb{P}^1$ parameterised by the associative Grassmannian, where the associative Grassmaniann is an 8-dimensional quaternion Kaehler manifold defined as the set of associative 3-planes in the 7-dimensional real vector space of the imaginary octonians. In the talk, we show that this story is quite analogous to the Penrose's twistor correspondence and that the geometric structures on the associative Grassmaniann nicely fit to this construction. This is a joint work with H. Hashimoto, K. Mashimo and M. Ohashi.

It is well known that the 6-dimensional sphere has a non-integrable almost complex structure which is introduced from the (right) multiplication of imaginary octonians. On this 6-sphere, there is a family of psuedo-holomorphic $\mathbb{C}\mathbb{P}^1$ parameterised by the associative Grassmannian, where the associative Grassmaniann is an 8-dimensional quaternion Kaehler manifold defined as the set of associative 3-planes in the 7-dimensional real vector space of the imaginary octonians. In the talk, we show that this story is quite analogous to the Penrose's twistor correspondence and that the geometric structures on the associative Grassmaniann nicely fit to this construction. This is a joint work with H. Hashimoto, K. Mashimo and M. Ohashi.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Extending Hori-Vafa toric mirror symmetry via SYZ and modular forms (English)

**Atsushi Kanazawa**(Harvard)Extending Hori-Vafa toric mirror symmetry via SYZ and modular forms (English)

[ Abstract ]

In this talk, I will introduce partial compactification for a class of toric Calabi-Yau manifolds. A fundamental question is how the Hori-Vafa toric mirror symmetry extends to this new class of Calabi-Yau manifolds. The answer leads us to a new connection between SYZ mirror symmetry and modular forms. If time permits, I will also discuss higher dimensional analogues of the Yau-Zaslow formula (for an elliptic K3 surface) in terms of Siegel modular forms. This talk is based on a joint work with Siu-Cheong Lau.

In this talk, I will introduce partial compactification for a class of toric Calabi-Yau manifolds. A fundamental question is how the Hori-Vafa toric mirror symmetry extends to this new class of Calabi-Yau manifolds. The answer leads us to a new connection between SYZ mirror symmetry and modular forms. If time permits, I will also discuss higher dimensional analogues of the Yau-Zaslow formula (for an elliptic K3 surface) in terms of Siegel modular forms. This talk is based on a joint work with Siu-Cheong Lau.

### 2015/12/09

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

K-theory in subfactors and conformal field theory

**David E. Evans**(Cardiff Univ.)K-theory in subfactors and conformal field theory

#### Number Theory Seminar

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Chern classes in Iwasawa theory (English)

**Ted Chinburg**(University of Pennsylvania & IHES)Chern classes in Iwasawa theory (English)

[ Abstract ]

Many of the main conjectures in Iwasawa theory can be phrased as saying that the first Chern class of an Iwasawa module is generated by a p-adic L-series. In this talk I will describe how higher Chern classes pertain to the higher codimension behavior of Iwasawa modules. I'll then describe a template for conjectures which would link such higher Chern classes to elements in the K-theory of Iwasawa algebras which are constructed from tuples of Katz p-adic L-series. I will finally describe an instance in which a result of this kind, for the second Chern class of an unramified Iwasawa module, can be proved over an imaginary quadratic field. This is joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. J. Taylor.

Many of the main conjectures in Iwasawa theory can be phrased as saying that the first Chern class of an Iwasawa module is generated by a p-adic L-series. In this talk I will describe how higher Chern classes pertain to the higher codimension behavior of Iwasawa modules. I'll then describe a template for conjectures which would link such higher Chern classes to elements in the K-theory of Iwasawa algebras which are constructed from tuples of Katz p-adic L-series. I will finally describe an instance in which a result of this kind, for the second Chern class of an unramified Iwasawa module, can be proved over an imaginary quadratic field. This is joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. J. Taylor.

### 2015/12/08

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)

**Yuichi Yamada**(The Univ. of Electro-Comm.)Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)

[ Abstract ]

The problem asking "Which knot yields a lens space by Dehn surgery" is

called "lens space surgery". Berge's list ('90) is believed to be the

complete list, but it is still unproved, even after some progress by

Heegaard Floer Homology.

This problem seems to enter a new aspect: study using 4-manifolds, lens

space surgery from lens spaces, checking hyperbolicity by computer.

In the talk, we review the structure of Berge's list and talk on our

study on pairs of distinct knots but yield same lens spaces, and

4-maniolds constructed from such pairs. This is joint work with Motoo

Tange (Tsukuba University).

The problem asking "Which knot yields a lens space by Dehn surgery" is

called "lens space surgery". Berge's list ('90) is believed to be the

complete list, but it is still unproved, even after some progress by

Heegaard Floer Homology.

This problem seems to enter a new aspect: study using 4-manifolds, lens

space surgery from lens spaces, checking hyperbolicity by computer.

In the talk, we review the structure of Berge's list and talk on our

study on pairs of distinct knots but yield same lens spaces, and

4-maniolds constructed from such pairs. This is joint work with Motoo

Tange (Tsukuba University).

### 2015/12/07

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Quenched invariance principle for random walks in time-dependent balanced random environment

**Jean-Dominique Deuschel**(TU Berlin)Quenched invariance principle for random walks in time-dependent balanced random environment

[ Abstract ]

We prove an almost sure functional limit theorem for a random walk in an space-time ergodic balanced environment under certain moment conditions. The proof is based on the maximal principle for parabolic difference operators. We also deal with the non-elliptic case, where the corresponding limiting diffusion matrix can be random in higher dimensions. This is a joint work with N. Berger, X. Guo and A. Ramirez.

We prove an almost sure functional limit theorem for a random walk in an space-time ergodic balanced environment under certain moment conditions. The proof is based on the maximal principle for parabolic difference operators. We also deal with the non-elliptic case, where the corresponding limiting diffusion matrix can be random in higher dimensions. This is a joint work with N. Berger, X. Guo and A. Ramirez.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Cycle connectivity and pseudoconcavity of flag domains (Japanese)

**Tatsuki Hayama**(Senshu Univ.)Cycle connectivity and pseudoconcavity of flag domains (Japanese)

[ Abstract ]

We consider an open real group orbit in a complex flag variety which has no non-constant function. We introduce Huckleberry's results on cycle connectivity and show that it is pseudoconcave if it satisfies a certain condition on the root system of the Lie algebra. In Hodge theory, we are mainly interested in the case where it is a Mumford-Tate domain. We also discuss Hodge theoretical meanings of this work.

We consider an open real group orbit in a complex flag variety which has no non-constant function. We introduce Huckleberry's results on cycle connectivity and show that it is pseudoconcave if it satisfies a certain condition on the root system of the Lie algebra. In Hodge theory, we are mainly interested in the case where it is a Mumford-Tate domain. We also discuss Hodge theoretical meanings of this work.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Flops and spherical functors (English)

**Alexey Bondal**(IPMU)Flops and spherical functors (English)

[ Abstract ]

I will describe various functors on derived categories of coherent sheaves

related to flops and relations between these functors. A categorical

version of deformation theory of systems of objects in abelian categories

will be outlined and its relation to flop spherical functors will be

presented.

I will describe various functors on derived categories of coherent sheaves

related to flops and relations between these functors. A categorical

version of deformation theory of systems of objects in abelian categories

will be outlined and its relation to flop spherical functors will be

presented.

### 2015/12/04

#### Colloquium

16:50-17:50 Room #123 (Graduate School of Math. Sci. Bldg.)

Exact forms and closed forms on some infinite product spaces appearing in the study of probability theory

(JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/sasada.html

**Makiko Sasada**(Graduate School of Mathematical Sciences, University of Tokyo)Exact forms and closed forms on some infinite product spaces appearing in the study of probability theory

(JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/sasada.html

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The q-Boson system and a deformation of the affine Hecke algebra (Japanese)

**Yoshihiro Takeyama**(Graduate School of Pure and Applied Sciences, University of Tsukuba)The q-Boson system and a deformation of the affine Hecke algebra (Japanese)

[ Abstract ]

The q-Boson system due to Sasamoto and Wadati is a one-dimensional "integrable" stochastic particle model. Its Q-matrix is constructed in the framework of the quantum inverse scattering method and we obtain the eigenvectors by means of the algebraic Bethe ansatz method. Recently it is found that the q-Boson model can be derived also from a representation of a deformation of the affine Hecke algebra and its representation. In this formulation we get the eigenvectors of the transpose of the Q-matrix which were constructed by the technique called the coordinate Bethe ansatz. In this talk I review the above results and discuss the relationship between the two methods.

The q-Boson system due to Sasamoto and Wadati is a one-dimensional "integrable" stochastic particle model. Its Q-matrix is constructed in the framework of the quantum inverse scattering method and we obtain the eigenvectors by means of the algebraic Bethe ansatz method. Recently it is found that the q-Boson model can be derived also from a representation of a deformation of the affine Hecke algebra and its representation. In this formulation we get the eigenvectors of the transpose of the Q-matrix which were constructed by the technique called the coordinate Bethe ansatz. In this talk I review the above results and discuss the relationship between the two methods.

### 2015/12/03

#### Seminar on Probability and Statistics

16:40-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Learning theory and sparsity ～ Sparsity and low rank matrix learning ～

**Arnak Dalalyan**(ENSAE ParisTech)Learning theory and sparsity ～ Sparsity and low rank matrix learning ～

[ Abstract ]

In this third lecture, we will present extensions of the previously introduced sparse recovery techniques to the problems of machine learning and statistics in which a large matrix should be learned from data. The analogue of the sparsity, in this context, is the low-rankness of the matrix. We will show that such matrices can be effectively learned by minimizing the empirical risk penalized by the nuclear norm. The resulting problem is a problem of semi-definite programming and can be solved efficiently even when the dimension is large. Theoretical guarantees for this method will be established in the case of matrix completion with known sampling distribution.

In this third lecture, we will present extensions of the previously introduced sparse recovery techniques to the problems of machine learning and statistics in which a large matrix should be learned from data. The analogue of the sparsity, in this context, is the low-rankness of the matrix. We will show that such matrices can be effectively learned by minimizing the empirical risk penalized by the nuclear norm. The resulting problem is a problem of semi-definite programming and can be solved efficiently even when the dimension is large. Theoretical guarantees for this method will be established in the case of matrix completion with known sampling distribution.

#### FMSP Lectures

16:40-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

(3)Sparsity and low rank matrix learning. (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

**Arnak Dalalyan**(ENSAE ParisTech)(3)Sparsity and low rank matrix learning. (ENGLISH)

[ Abstract ]

In this third lecture, we will present extensions of the previously introduced sparse recovery techniques to the problems of machine learning and statistics in which a large matrix should be learned from data. The analogue of the sparsity, in this context, is the low-rankness of the matrix. We will show that such matrices can be effectively learned by minimizing the empirical risk penalized by the nuclear norm. The resulting problem is a problem of semi-definite programming and can be solved efficiently even when the dimension is large. Theoretical guarantees for this method will be established in the case of matrix completion with known sampling distribution.

[ Reference URL ]In this third lecture, we will present extensions of the previously introduced sparse recovery techniques to the problems of machine learning and statistics in which a large matrix should be learned from data. The analogue of the sparsity, in this context, is the low-rankness of the matrix. We will show that such matrices can be effectively learned by minimizing the empirical risk penalized by the nuclear norm. The resulting problem is a problem of semi-definite programming and can be solved efficiently even when the dimension is large. Theoretical guarantees for this method will be established in the case of matrix completion with known sampling distribution.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

### 2015/12/02

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Drinfeld center and representation theory for monoidal categories

**Makoto Yamashita**(Ochanomizu Univ.)Drinfeld center and representation theory for monoidal categories

#### Seminar on Probability and Statistics

14:55-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Learning theory and sparsity ～ Lasso, Dantzig selector and their statistical properties ～

**Arnak Dalalyan**(ENSAE ParisTech)Learning theory and sparsity ～ Lasso, Dantzig selector and their statistical properties ～

[ Abstract ]

In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

#### FMSP Lectures

14:55-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

(2)Lasso, Dantzig selector and their statistical properties. (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

**Arnak Dalalyan**(ENSAE ParisTech)(2)Lasso, Dantzig selector and their statistical properties. (ENGLISH)

[ Abstract ]

In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

[ Reference URL ]In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

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