I am engaged in research to develop machine learning theory based on information geometry. My background in master’s course was physics. So I am intensely interested in the interdisciplinary academic field of intelligent informatics and physics. When I can describe things in machine learning with physics terms, I am thrilled.


Many-body approximation for tensors

Tensor factorization has fundamental difficulties in rank tuning and optimization. To avoid these difficulties, we develop a rank-free energy-based tensor factorization, called many-body approximation, that allows intuitive modeling of tensors and global optimization. Our approach models tensors as distributions via the energy function, which describes interactions between modes, and a dually flat statistical manifold is induced.

Related papers
K.Ghalamkari, et al., “Many-body Approximation for Non-negative Tensors” presented in NeurIPS2023.

Faster rank-1 NMF with missing values

Nonnegative multiple matrix factorization (NMMF) is the task of decomposing multiple nonnegative matrices by shared factors. We have derived a closed formula of the best rank-1 NMMF when the cost function is defined by KL divergence. Using this solution formula, we proposed A1GM, a faster method to find approximate solutions of rank-1 NMF when the input matrix has missing values.


Related papers
K.Ghalamkari, et al., “Fast Rank-1 NMF for Missing Data with KL Divergence” presented in AISTATS2022.
K.Ghalamkari, et al., “Non-negative low-rank approximations for multi-dimensional arrays on statistical manifold” pubulished in Information Geometry (Springer)

Tensor low-rank approximation

Tensor low-rank approximation is the task of approximating a tensor (multidimensional array) with a low-rank tensor. To date, most low-rank approximation methods are based on gradient descent using the derivative of the cost function, which often requires careful tuning of initialization, learning rate, and a tolerance threshold. By mapping tensors to probability distributions and applying the theory of projection in information geometry, we have developed Legendre Tucker Rank Reduction (LTR), a fast non-negative low-rank approximation method that is not based on the gradient method.

From the viewpoint of information geometry, we also explain that we can regard the rank-1 approximation as a mean-field approximation because the probability distribution corresponding to the rank-1 tensor is expressed as a product of independent distributions.

テンソル(多次元配列)をランクの低いテンソルで近似するタスクがテンソル低ランク近似です.従来,この低ランク近似は,入力テンソルと低ランク近似後のテンソルの誤差を小さくする勾配法によって行われてきました.しかし,勾配法に基づく手法は,適切な初期値や収束判定,学習率の設定などのチューニングが必要です.私たちは,テンソルを確率分布と対応付け,情報幾何学の射影の理論を活用することで,勾配法に基づかない非負テンソルの高速な低ランク近似手法Legendre Tucker Rank Reduction(LTR)を開発しました.


Related papers
K.Ghalamkari, et al., “Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation” presented in NeurIPS2021.
K.Ghalamkari, et al., “Non-negative low-rank approximations for multi-dimensional arrays on statistical manifold” pubulished in Information Geometry (Springer)

Valley polarization in layer material by light absorption

I received my master’s degree in theoretical condensed matter physics in hexagonal lattice systems such as graphene, h-BN, Silicene, and Stanene.

Hexagonal lattice systems, in which two atoms in a unit cell are different atoms, have K valleys and K’ valleys in the Brillouin zone. It is known that the electrons in the K(K’) valleys are excited when left (right) circularly polarized light is incident in this system. The system is said to be valley polarized when there is a bias in which valley electrons are more likely to be excited depending on the polarization state of the irradiated light. I have analytically demonstrated valley polarization in a hexagonal lattice system consisting of atoms with complex electronic orbitals such as TMDs [1]. We also provide an analytical explanation of how the valley polarization varies with the energy band gap and incident energy [2].

Furthermore, I have clarified the relationship between band inversion and valley polarization by studying the presence or absence of valley polarization in the Haldane model [3].




[1] Y.Tatsumi, et al. Laser energy dependence of valley polarization in transition-metal dichalcogenides, Phys. Rev. B 94, 235408 (2016) 
[2] K.Ghalamkari, et al. Energy Band Gap Dependence of Valley Polarization of the Hexagonal Lattice, J. Phys. Soc. Jpn. 87, 024710 (2018)
[3] K.Ghalamkari, et al. Perfect circular dichroism in the Haldane model, J. Phys. Soc. Jpn. 87, 063708 (2018)