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Shu Nakamura

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Published work

15 published item(s)

preprint2026arXiv

Action Motifs: Self-Supervised Hierarchical Representation of Human Body Movements

Effective human behavior modeling requires a representation of the human body movement that capitalizes on its compositionality. We propose a hierarchical representation consisting of Action Atoms that capture the atomic joint movements and Action Motifs that are formed by their temporal compositions and encode similar body movements found across different overall human actions. We derive A4Mer, a nested latent Transformer to learn this hierarchical representation from human pose data in a fully self-supervised manner. A4Mer splits a 3D pose sequence into variable-length segments and represents each segment as a single latent token (Action Atoms). Through bottom-up representation learning, temporal patterns composed of these Action Atoms, which capture meaningful temporal spans of reusable, semantic segments of body movements, naturally emerge (Action Motifs). A4Mer achieves this with a unified pretext task of masked token prediction in their respective latent spaces. We also introduce Action Motif Dataset (AMD), a large-scale dataset of multi-view human behavior videos with full SMPL annotations. We introduce a novel use of cameras by mounting them on the feet to achieve their frame-wise annotations despite frequent and heavy body occlusions. Experimental results demonstrate the effectiveness of A4Mer for extracting meaningful Action Motifs, which significantly benefit human behavior modeling tasks including action recognition, motion prediction, and motion interpolation.

preprint2022arXiv

Continuum limit of the lattice quantum graph Hamiltonian

We consider the quantum graph Hamiltonian on the square lattice in Euclidean space, and we show that the spectrum of the Hamiltonian converges to the corresponding Schrödinger operator on the Euclidean space in the continuum limit, and that the corresponding eigenfunctions and eigenprojections also converge in some sense. We employ the discrete Schrödinger operator as the intermediate operator, and we use a recent result by the second and third author on the continuum limit of the discrete Schrödinger operator.

preprint2021arXiv

Quantization optimized with respect to the Haar basis

We propose a method of data quantization of finite discrete-time signals which optimizes the error estimate of low frequency Haar coefficients. We also discuss the error/noise bounds of this quantization in the Fourier space. Our result shows one can quantize any discrete-time analog signal with high precision at low frequencies. Our method is deterministic, and it employs no statistical arguments, nor any probabilistic assumptions.

preprint2020arXiv

Remarks on scattering matrices for Schrödinger operators with critically long-range perturbations

We consider scattering matrix for Schrödinger-type operators on $\mathbb{R}^d$ with perturbation $V(x)=O(\langle x\rangle^{-1})$ as $|x|\to\infty$. We show that the scattering matrix (with time-independent modifiers) is a pseudodifferential operator, and analyze its spectrum. We present examples of which the spectrum of the scattering matrices have dense point spectrum, and absolutely continuous spectrum, respectively.

preprint2016arXiv

Microlocal resolvent estimates, revisited

Let $H$ be a Schrödinger type operator with long-range perturbation. We study the wave front set of the distribution kernel of $(H-λ\mp i0)^{-1}$, where $λ$ is in the absolutely continous spectrumof $H$.The result is a refinement of the microlocal resolvent estimate of Isozaki-Kitada \cite{IK1,IK2}. We prove the result for a class of pseudodifferential operators on manifolds so that they apply to discrete Schrödinger operators and higher order operators on the Euclidean space. The proof relies on propagation estimates, whereas the original proof of Isozaki-Kitada relies on a construction of parametrices.

preprint2016arXiv

Spectral shift functions and Dirichlet-to-Neumann maps

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schrödinger operators with compactly supported potentials, and finally, Schrödinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.

preprint2014arXiv

Microlocal properties of scattering matrices

We consider scattering theory for a pair of operators $H_0$ and $H=H_0+V$ on $L^2(M,m)$, where $M$ is a Riemannian manifold, $H_0$ is a multiplication operator on $M$ and $V$ is a pseudodifferential operator of order $-μ$, $μ>1$. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.

preprint2013arXiv

Propagation of singularities for Schrödinger equations with modestly long range type potentials

In a previous paper by the second author, we discussed a characterization of the microlocal singularities for solutions to Schrödinger equations with long range type perturbations, using solutions to a Hamilton-Jacobi equation. In this paper we show that we may use Dollard type approximate solutions to the Hamilton-Jacobi equation if the perturbation satisfies somewhat stronger conditions. As applications, we describe the propagation of microlocal singularities for $e^{itH_0}e^{-itH}$ when the potential is asymptotically homogeneous as $|x|\to\infty$, where $H$ is our Schrödinger operator, and $H_0$ is the free Schrödinger operator, i.e., $H_0=-\frac12 \triangle$. We show $e^{itH_0}e^{-itH}$ shifts the wave front set if the potential $V$ is asymptotically homogeneous of order 1, whereas $e^{itH}e^{-itH_0}$ is smoothing if $V$ is asymptotically homogenous of order $β\in (1,3/2)$.

preprint2012arXiv

Microlocal properties of scattering matrices for Schrödinger equations on scattering manifolds

Let $M$ be a scattering manifold, i.e., a Riemannian manifold with asymptotically conic structure, and let $H$ be a Schrödinger operator on $M$. We can construct a natural time-dependent scattering theory for $H$ with a suitable reference system, and the scattering matrix is defined accordingly. We here show the scattering matrices are Fourier integral operators associated to a canonical transform on the boundary manifold generated by the geodesic flow. In particular, we learn that the wave front sets are mapped according to the canonical transform. These results are generalizations of a theorem by Melrose and Zworski, but the framework and the proof are quite different. These results may be considered as generalizations or refinements of the classical off-diagonal smoothness of the scattering matrix for 2-body quantum scattering on Euclidean spaces.

preprint2012arXiv

The Spectral Shift Function and The Friedel Sum Rule

We study the relationship between the spectral shift function and the excess charge in potential scattering theory. Although these quantities are closely related to each other, they have been often formulated in different settings so far. Here we first give an alternative construction of the spectral shift function, and then we prove that the spectral shift function thus constructed yields the Friedel sum rule.

preprint2012arXiv

The spectrum of the scattering matrix near resonant energies in the semiclassical limit

The object of study in this paper is the on-shell scattering matrix $S(E)$ of the Schrödinger operator with the potential satisfying assumptions typical in the theory of shape resonances. We study the spectrum of $S(E)$ in the semiclassical limit when the energy parameter $E$ varies from $E_\text{res}-\varepsilon$ to $E_\text{res}+\varepsilon$, where $E_\text{res}$ is a real part of a resonance, and $\varepsilon$ is sufficiently small. The main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle. This result is closely related to the Breit-Wigner effect.

preprint2008arXiv

Time-dependent scattering theory for Schrödinger operators on scattering manifolds

We construct a time-dependent scattering theory for Schrödinger operators on a manifold $M$ with asymptotically conic structure. We use the two-space scattering theory formalism, and a reference operator on a space of the form $R\times \partial M$, where $\partial M$ is the boundary of $M$ at infinity. We prove the existence and the completeness of the wave operators, and show that our scattering matrix is equivalent to the absolute scattering matrix, which is defined in terms of the asymptotic expansion of generalized eigenfunctions. Our method is functional analytic, and we use no microlocal analysis in this paper.

preprint2004arXiv

Bounds on the spectral shift function and the density of states

We study spectra of Schrödinger operators on $\RR^d$. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values $μ_n$ of the difference of the semigroups as $n\to \infty$ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger operators. The single site potential $u$ is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.