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We showed that the strong Sarkisov Program of dimension $d$ can be derived from termination of specific log flips in dimension $\leq d-1$. As a corollary, we show that the strong Sarkisov Program holds in dimension 4. Additionally, we prove the weak Sarkisov program with decreasing (augmented) Sarkisov degree. Finally, using the theory of syzygies of Mori fibre spaces, we obtain a directed diagram that encodes all the information of the strong Sarkisov Program.

We introduce homological and homotopical $r$-syzygies of Mori fibre spaces as a generalization of Sarkisov links and relations of Sarkisov links. For any proper morphism $Y/R$, we construct a contractible (if not empty) CW complex such that there is a 1-1 correspondence between its cells and the central models of $Y/R$. We derive from this CW complex a long exact sequence and a spectral sequence converging to the (co)homology of the relative birational automorphism group of $Y/R$. As an application, we compute the second group homology of the second Cremona group.

Dataset condensation is a crucial tool for enhancing training efficiency by reducing the size of the training dataset, particularly in on-device scenarios. However, these scenarios have two significant challenges: 1) the varying computational resources available on the devices require a dataset size different from the pre-defined condensed dataset, and 2) the limited computational resources often preclude the possibility of conducting additional condensation processes. We introduce You Only Condense Once (YOCO) to overcome these limitations. On top of one condensed dataset, YOCO produces smaller condensed datasets with two embarrassingly simple dataset pruning rules: Low LBPE Score and Balanced Construction. YOCO offers two key advantages: 1) it can flexibly resize the dataset to fit varying computational constraints, and 2) it eliminates the need for extra condensation processes, which can be computationally prohibitive. Experiments validate our findings on networks including ConvNet, ResNet and DenseNet, and datasets including CIFAR-10, CIFAR-100 and ImageNet. For example, our YOCO surpassed various dataset condensation and dataset pruning methods on CIFAR-10 with ten Images Per Class (IPC), achieving 6.98-8.89% and 6.31-23.92% accuracy gains, respectively. The code is available at: https://github.com/he-y/you-only-condense-once., Comment: Accepted by NeurIPS 2023

While one-dimensional convolutional neural networks (1D-CNNs) have been empirically proven effective in time series classification tasks, we find that there remain undesirable outcomes that could arise in their application, motivating us to further investigate and understand their underlying mechanisms. In this work, we propose a Temporal Convolutional Explorer (TCE) to empirically explore the learning behavior of 1D-CNNs from the perspective of the frequency domain. Our TCE analysis highlights that deeper 1D-CNNs tend to distract the focus from the low-frequency components leading to the accuracy degradation phenomenon, and the disturbing convolution is the driving factor. Then, we leverage our findings to the practical application and propose a regulatory framework, which can easily be integrated into existing 1D-CNNs. It aims to rectify the suboptimal learning behavior by enabling the network to selectively bypass the specified disturbing convolutions. Finally, through comprehensive experiments on widely-used UCR, UEA, and UCI benchmarks, we demonstrate that 1) TCE's insight into 1D-CNN's learning behavior; 2) our regulatory framework enables state-of-the-art 1D-CNNs to get improved performances with less consumption of memory and computational overhead., Comment: Accepted as CIKM'23 Long Paper

We propose a machine learning approach to study topological quantities related to the Sasakian and $G_2$-geometries of contact Calabi-Yau $7$-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-N\"ordstrom invariant of the natural $G_2$-structure of the $7$-dimensional link of a weighted projective Calabi-Yau $3$-fold hypersurface singularity, for 7549 of the 7555 possible $\mathbb{P}^4(\textbf{w})$ projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the $\mathbb{P}^4(\textbf{w})$ weights alone, using both neural networks and a symbolic regressor which achieve $R^2$ scores of 0.969 and 0.993 respectively. Additionally, properties of the respective Gr\"obner bases are well-learnt, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further induced novel conjectures to be raised., Comment: 10 pages; 5 figures; 1 table

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for $A_8$, $D_8$ and $E_8$ for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output -- the invariants -- is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results., Comment: v1: 34 pages, 16 Figures, 12 Tables. v2: Typos corrected and some comments added. Matches the author-accepted version for publication in Advances in Applied Clifford Algebras

In recent years, graph neural networks (GNN) have achieved significant developments in a variety of graph analytical tasks. Nevertheless, GNN's superior performance will suffer from serious damage when the collected node features or structure relationships are partially missing owning to numerous unpredictable factors. Recently emerged graph completion learning (GCL) has received increasing attention, which aims to reconstruct the missing node features or structure relationships under the guidance of a specifically supervised task. Although these proposed GCL methods have made great success, they still exist the following problems: the reliance on labels, the bias of the reconstructed node features and structure relationships. Besides, the generalization ability of the existing GCL still faces a huge challenge when both collected node features and structure relationships are partially missing at the same time. To solve the above issues, we propose a more general GCL framework with the aid of self-supervised learning for improving the task performance of the existing GNN variants on graphs with features and structure missing, termed unsupervised GCL (UGCL). Specifically, to avoid the mismatch between missing node features and structure during the message-passing process of GNN, we separate the feature reconstruction and structure reconstruction and design its personalized model in turn. Then, a dual contrastive loss on the structure level and feature level is introduced to maximize the mutual information of node representations from feature reconstructing and structure reconstructing paths for providing more supervision signals. Finally, the reconstructed node features and structure can be applied to the downstream node classification task. Extensive experiments on eight datasets, three GNN variants and five missing rates demonstrate the effectiveness of our proposed method., Comment: Accepted by 23rd IEEE International Conference on Data Mining (ICDM 2023)

Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi-Yau four-folds. In some instances, the Hodge numbers we compute are new as well., Comment: 24 pages, 2 figures. Accepted version

We argue that one can associate a pseudo-time with sequences of configurations generated in the course of classical Monte Carlo simulations for a single-minimum bound state, if the sampling is optimal. Hereby the sampling rates can be, under special circumstances, calibrated against the relaxation rate and frequency of motion of an actual physical system. The latter possibility is linked to the optimal sampling regime being a universal crossover separating two distinct suboptimal sampling regimes analogous to the physical phenomena of diffusion and effusion, respectively. Bound states break symmetry; one may thus regard the pseudo-time as a quantity emerging together with the bound state. Conversely, when transport among distinct bound states takes place -- thus restoring symmetry -- a pseudo-time can no longer be defined. One can still quantify activation barriers, if the latter barriers are smooth, but the simulation becomes impractically slow and pertains to overdamped transport only. Specially designed Monte Carlo moves that bypass activation barriers -- so as to accelerate sampling of the thermodynamics -- amount to effusive transport and lead to severe under-sampling of transition-state configurations that separate distinct bound states while destroying the said universality. Implications of the present findings for simulations of glassy liquids are discussed.

We give the first numerical calculation of the spectrum of the Laplacian acting on bundle-valued forms on a Calabi-Yau three-fold. Specifically, we show how to compute the approximate eigenvalues and eigenmodes of the Dolbeault Laplacian acting on bundle-valued $(p,q)$-forms on K\"ahler manifolds. We restrict our attention to line bundles over complex projective space and Calabi-Yau hypersurfaces therein. We give three examples. For two of these, $\mathbb{P}^3$ and a Calabi-Yau one-fold (a torus), we compare our numerics with exact results available in the literature and find complete agreement. For the third example, the Fermat quintic three-fold, there are no known analytic results, so our numerical calculations are the first of their kind. The resulting spectra pass a number of non-trivial checks that arise from Serre duality and the Hodge decomposition. The outputs of our algorithm include all the ingredients one needs to compute physical Yukawa couplings in string compactifications., Comment: 52 pages, 6 figures; v2 - corrected typo in quintic equation

A homage to the life and mathematics of John K. S. McKay. Obituary for the Bulletin of the London Mathematical Society., Comment: 20 pages, 6 figures, invited by the Bulletin of the LMS

We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and unsupervised machine learning have found surprising success. We thus advocate the programme of machine learning mathematical structures, and formulating conjectures via pattern recognition, in other words using artificial intelligence to help one do mathematics. This is an invited chapter contribution to Elsevier's Handbook of Statistics, Volume 49: Artificial Intelligence edited by S.~G.~Krantz, A.~S.~R.~Srinivasa Rao, and C.~R.~Rao.

The remarkable performance of deep Convolutional neural networks (CNNs) is generally attributed to their deeper and wider architectures, which can come with significant computational costs. Pruning neural networks has thus gained interest since it effectively lowers storage and computational costs. In contrast to weight pruning, which results in unstructured models, structured pruning provides the benefit of realistic acceleration by producing models that are friendly to hardware implementation. The special requirements of structured pruning have led to the discovery of numerous new challenges and the development of innovative solutions. This article surveys the recent progress towards structured pruning of deep CNNs. We summarize and compare the state-of-the-art structured pruning techniques with respect to filter ranking methods, regularization methods, dynamic execution, neural architecture search, the lottery ticket hypothesis, and the applications of pruning. While discussing structured pruning algorithms, we briefly introduce the unstructured pruning counterpart to emphasize their differences. Furthermore, we provide insights into potential research opportunities in the field of structured pruning. A curated list of neural network pruning papers can be found at https://github.com/he-y/Awesome-Pruning, Comment: Work in progress