## 出版物

### MIプレプリント

Title：Selection of classification boundaries using the logistic regression

Author : Hidetoshi Matsui

Abstract: We propose the method for selecting classification boundaries for the logistic regression model, by applying the sparse regularization. We can investigate which combination of classification boundaries are truly necessary for the multinomial logistic model by encouraging some of coefficient parameters themselves or their differences toward exactly zeros. The model is estimated by the maximum penalized likelihood method with the fused lasso type penalty. We also introduce some model selection criteria for evaluating models estimated by the penalized likelihood method. Simulation and real data analysis show insights that the proposed method goes well.

accepted by Bulletin of Informatics and Cybernetics on December 2015,

and should be available online three years later.

https://www.lib.kyushu-u.ac.jp/ja/publications_kyushu/bic

#### 2013-15

Title：The Fault-Tolerant Facility Location Problem with Submodular Penalties

Author : Naoyuki Kamiyama

Abstract: In this paper, we consider the fault-tolerant facility location problem with submodular penal- ties that is a common generalization of the fault-tolerant facility location problem and the facility location problem with submodular penalties. For this problem, we present a combinatorial 3 • HR -approximation algorithm, where R is the maximum connectivity requirement and HR is the R-th harmonic number.

Our algorithm is a common generalization of the algorithm for the the fault-tolerant facility location problem presented by Jain and Vazirani (2003) and that for the facility location problem with submodular penalties presented by Du, Lu and Xu (2012).

#### 2013-14

Title：Variable selection for functional linear models with functional predictors and a functional response

Author : Hidetoshi Matsui

Abstract: We consider a variable selection problem for functional linear models where both multiple predictors and a response are functions. Especially we assume that variables are given as functions of time and then construct the historical functional linear model which takes the relationship of dependences of predictors and a response into consideration. Unknown parameters included in the model are estimated by the maximum penalized likelihood method with the L1 penalty. We can simultaneously estimate and select variables given as functions using the L1 type penalty. A regularization parameter involved in the regularization method is decided by a model selection criterion. The effectiveness of the proposed method is investigated by simulation studies and real data analysis.

#### 2013-13

Title：The First Painleve Equation on the Weighted Projective Space

Author : Hayato Chiba

Abstract: The first Painleve equation is expressed as a vector field on the three dimensional weighted projective space CP^3(3,2,4,5).

With the aid of the dynamical systems theory and the orbifold structure of CP^3(3,2,4,5), a simple proof of the Painleve property and the construction of the space of initial conditions are given.

In particular, Painleve's transformation is geometrically derived, which proves to be a Darboux coordinates of a certain algebraic surface with a holomorphic symplectic form.

#### 2013-12

Title：Predictive model selection criteria for Bayesian lasso

Author : Shuichi Kawano, Ibuki Hoshina, Kazuki Matsuda & Sadanori Konishi

Abstract: We consider the Bayesian lasso for regression, which is an L1 penalized regression based on Bayesian approach. In Bayesian theory, a crucial issue is the specification of prior distributions for parameters, which leads to the selection of values of hyperparameters included in the prior distributions. In order to select the values of the hyperparameters, we introduce a model selection criterion by evaluating the Bayesian predictive distribution for the Bayesian lasso. Several numerical studies are presented to illustrate the effectiveness of our proposed modeling procedure.

Journal of the Japanese Society of Computational Statistics

Vol. 28 (2015) No. 1 p. 67-82

http://dx.doi.org/10.5183/jjscs.1501001_220

#### 2013-11

Title：Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations

Author : Masatoshi Okita

In this paper we are concerned with the convergence rates for the compressible Navier-Stokes equations in the whole space. It is proved that the perturbations decay in critical spaces, if the initial perturbations are small in critical space.

#### 2013-10

Title：Equivalence between the eigenvalue problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun differential equations, eigenstates degeneration and Rabi model

Author : Masato Wakayama

Abstract: The initial aim of the present paper is to provide a complete description of the eigenvalue problem of the non-commutative harmonic oscillator (NcHO), which is defined by a (two-by-two) matrix-valued self-adjoint parity-preserving ordinary differential operator [21], in terms of Heun's ordinary differential equations, the second order Fuchsian differential equations with four regular singularities in a complex domain. This description has been achieved for odd eigenfunctions in Ochiai [18] nicely but missing up to now for the even parity, which is more important from the viewpoint of determination of the ground state. As a by-product of this study, using the monodromy representation of Heun's equation, we prove that the multiplicity of the eigenvalue of the NcHO is at most two. Moreover, we give a condition for the existence of a finite-type eigenfunction (given by essentially a finite sum of Hermite functions) for the eigenvalue problem and an explicit example of such eigenvalues, from which one finds that doubly degenerate eigenstates of the NcHO actually exist even in the same parity. In the final section, as the second main purpose of this paper, we discuss a connection between the quantum Rabi model [2, 15, 32] and the element of the universal enveloping algebra u(sl_2) of Lie algebra sl_2 naturally arising from the NcHO through the oscillator representation. Precisely, an equivalent picture of the Rabi model drawn by a confluent Heun equation is obtained from the Heun operator defined by that element in u(sl_2) under a (flat picture of non-unitary) principal series representation of sl_2.

#### 2013-9(Published)

Title：Packing Arborescences in Acyclic Temporal Networks

Author : Naoyuki Kamiyama

Abstract: A temporal network is a finite directed graph in which each arc has a time label specifying the time at which its end-vertices communicate. An arborescence in a temporal network is said to be time-respecting, if the time labels on every directed path from the root in this arborescence are monotonically non-decreasing. In this paper, we consider the problem of packing time-respecting arborescences in a temporal network. Precisely speaking, we study an extension of Edmonds' arc-disjoint arborescences theorem in a temporal network. Unfortunately, it is known that a natural extension of Edmonds' arc-disjoint arborescences theorem in a temporal network does not hold. In this paper, we first show that this extension does not hold, even if an input temporal network is acyclic. Next, we prove that if an input temporal network is acyclic and pre-flow, this extension holds and we can find arc-disjoint time-respecting arborescences in polynomial time. Furthermore, we extend our main result to the problem of packing time-respecting partial arborescences.

Information Processing Letters

#### 2013-8(Published)

Title：Variable selection for varying coefficient models with the sparse regularization

Author : Hidetoshi Matsui & Toshihiro Misumi

Abstract: Varying-coefficient models are useful tools for analyzing longitudinal data. They can effectively describe a relationship between predictors and responses repeatedly measured. We consider the problem of selecting variables in the varying-coefficient models via the adaptive elastic net regularization. Coefficients given as functions are expressed by basis expansions, and then parameters involved in the model are estimated by the penalized likelihood method using the coordinate descent algorithm derived for solving the problem of sparse regularization. We examine the effectiveness of our modeling procedure through Monte Carlo simulations and real data analysis.

Computational Statistics

#### 2013-7(Published)

Title：Variable and boundary selection for functional data via multiclass logistic regression modeling

Author : Hidetoshi Matsui

Abstract: L1 penalties such as the lasso provide solutions with some coefficients to be exactly zeros, which lead to variable selection in regression settings. They also can select variables which affect the classification by being applied to the logistic regression model. We focus on the form of L1 penalties in logistic regression models for functional data, especially in the case for classifying the functions into three or more groups. We provide penalties that appropriately select variables in the functional multinomial regression modeling. Simulation and real data analysis show that we should select the form of the penalty in accordance with the purpose of the analysis.

Computational Statistics and Data Analysis

#### 2013-6

Title：Improved bounds on Restricted isometry for compressed sensing

Author : Hiroshi Inoue

Abstract: This paper discusses new bounds for restricted isometry property in compressed sensing. In the literature, E.J. Candes has proved that delta_{2s} < sqrt{2}-1 is a sufficient condition via l_{1} optimization having s-sparse vector solution. Later, many researchers have improved the sufficient conditions on delta_{2s} or delta_{s}. In this paper, we have improved the sufficient condition to delta_{s}<0.309 and have given the sufficient condition to delta_{k}(s<k) using an idea of Q. Mo and S. Li' result. Furthermore, we have improved the sufficient conditions to delta_{2s}<0.593 and delta_{s}<0.472 in special case.

#### 2013-5

Title：RIPless Theory for Compressed Sensing

Author : Hiroshi Inoue

Abstract: This paper discusses the theory for RIPless in compressed sensing (CS). In the literature, E.J. Candes has proved that delta_{2s} < sqrt{2}-1 is a sufficient condition via l_{1} optimization having s-sparse vector solution. Later, many researchers have improved the sufficient conditions on delta_{2s} or delta_{s}. Above researches have supposed that a matrix A obeys RIP and a signal to recover is sparse. In this paper, we do not suppose that a matrix A obeys RIP and a signal is sparse. We propose the RIPless theory and the method of any signal recovery for noiseless and noisy cases in CS.

#### 2013-4

Title：On Counting Output Patterns of Logic Circuits

Author : Naoyuki Kamiyama

Abstract. In this paper, we consider the problem of counting output patterns of a circuit with gates having fan-in 2. For the case where every gate computes the same Boolean function f, Uchizawa, Wang, Morizumi and Zhou (2013) proved that the problem is solvable in polynomial time if f is the parity function or any degenerate function, while this problem is #P-complete if f is one of the other functions. In this paper, we extend the positive result of Uchizawa, Wang, Morizumi and Zhou (2013) to the case where gates may compute different Boolean functions.

#### 2013-3

Title：Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes

Author : Hiroki Masuda

Abstract: The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit statistics for goodness of fit tests, consistent against ``presence of a jump component'' and ``diffusion-coefficient misspecification''; then, the acceptance of the null hypothesis may serve as a collateral evidence for using the diffusion type model. Especially, our test procedure clarifies how to remove the bias caused by plugging in a diffusion-coefficient estimator.

#### 2013-2

Title：A unified view of topological invariants of barotropic and baroclinic fluids and their application to formal stability analysis of three-dimensional ideal gas flows

Author : Yasuhide Fukumoto & Hirofumi Sakuma

Abstract: Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. The similar is shown to be true of a baroclinic fluid. A cross-helicity representation is given to the Casimir invariant, a class of integrals including an arbitrary function of the specific entropy and the potential vorticity. We then develop a new energy-Casimir convexity method for three-dimensional stability of equilibria of general rotating flows of an ideal baroclinic gas, without appealing to the Boussinesq approximation. By fully exploiting the Casimir invariant, we have succeeded in ruling out a term including the gradient of a dependent variable from the energy-Casimir function and have established a sharp linear stability criterion, being an extension of the Richardson-number criterion.

#### 2013-1

Title：A Counter-example to Thomson-Tait-Chetayev's Theorem

Author : Abuduwaili Paerhati & Yasuhide Fukumoto

Abstract: A counter-example is given to Thomson-Tait-Chetayev's (TTC) theorem which states that, for a system with an unstable potential, a state stabilized by gyroscopic forces goes unstable after the addition of arbitrary dissipation. The example is brought by a system, closely related with a heavy symmetrical top, describing motion of a charged spherical pendulum subjected to the Lorentz force, in a magnetic-monopole field sitting at the sphere center, as well as the gravity force. A drag force proportional to the velocity is exerted on the pendulum. The upright state, an equilibrium stabilized by the Lorentz force, is shown to be exempted from the TTC theorem.

The numerical calculation of the full nonlinear system is performed for precession. For the slow precession, the drag force acts to continuously tilt down the top axis toward vertically downward equilibrium, following the dissipation-induced instability. On the contrary, for the fast precession, the drag acts to continuously tilt up the axis against the gravity force, despite losing the energy.

## 2013-16(Accepted)