Daily Papers

We develop a technique to perform efficient unitary time evolution of a general spin Hamiltonian based on the system with identical qubits, as naturally realized by permutation symmetric atomic ensembles. We employ Cartan decomposition method to the associated Hamiltonian Lie algebra and construct the symmetry adapted generators to implement unitary operator of the ensemble Hamiltonian. This enables the compact circuit representations of the system dynamics in terms of experimentally accessible collective spin operations and exploits the Dicke basis structure of the ensembles as an effective qudit representation. We benchmark Cartan based evolution against both exact time evolution and the Suzuki Trotter product formula. Our results show that the Cartan protocol accurately reproduces the operator dynamics, spin expectation values, and fluctuation observables with zero error that are substantially smaller than those obtained through Trotterization while maintaining an optimized circuit structure. In addition, scaling analysis with the ensemble size demonstrates that the Cartan error remains independent of the atom number while the corresponding Trotter error increases with the system size, highlighting the improved efficiency of the Cartan approach for a larger system size.

The cosmology of the F(R) gravity rebuilding by the Cartan formalism is investigated. This is called Cartan F(R) gravity. The well-known F(R) gravity has been introduced to extend the standard cosmology, e.g. to explain the cosmological accelerated expansion as the inflation. Cartan F(R) gravity is based on the Riemann-Cartan geometry. The curvature R can separate to two parts, one is derived from the Levi-Civita connection and the other from the torsion. Assuming the matter-independent spin connection, we have successfully rewritten the action of Cartan F(R) gravity into the Einstein-Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. This feature simplifies building and analysis of new model of inflation. In this paper, we study two models, the power-law model and logarithmic model, and evaluate fluctuations in the cosmological microwave background (CMB) radiation. We found the robustness of CMB fluctuation by the analytical computation and confirm this feature by the numerical calculation.

A complete theory of null Cartan normal helices in Minkowski space-time E^4_1 is developed. Two algebraic conditions, obtained by successive differentiation of the helix invariant along a unit C-constant normal field, fully characterize null Cartan helices; the quadratic condition yields two mutually orthogonal helix axes in the Lorentzian metric. Special field types are analyzed and null Cartan cubics are shown to be normal helices. On a timelike hypersurface, a Darboux frame with six curvature functions is constructed from first principles, the normal isophotic condition is shown to reduce to a linear first-order ODE, and the existence of normal silhouettes in E^4_1 is established.

In time series forecasting, effectively disentangling intricate temporal patterns is crucial. While recent works endeavor to combine decomposition techniques with deep learning, multiple frequencies may still be mixed in the decomposed components, e.g., trend and seasonal. Furthermore, frequency domain analysis methods, e.g., Fourier and wavelet transforms, have limitations in resolution in the time domain and adaptability. In this paper, we propose D-PAD, a deep-shallow multi-frequency patterns disentangling neural network for time series forecasting. Specifically, a multi-component decomposing (MCD) block is introduced to decompose the series into components with different frequency ranges, corresponding to the "shallow" aspect. A decomposition-reconstruction-decomposition (D-R-D) module is proposed to progressively extract the information of frequencies mixed in the components, corresponding to the "deep" aspect. After that, an interaction and fusion (IF) module is used to further analyze the components. Extensive experiments on seven real-world datasets demonstrate that D-PAD achieves the state-of-the-art performance, outperforming the best baseline by an average of 9.48% and 7.15% in MSE and MAE, respectively.

In diffraction-based crystal structure analysis, thermal ellipsoids, quantified via Anisotropic Displacement Parameters (ADPs), are critical yet challenging to determine. ADPs capture atomic vibrations, reflecting thermal and structural properties, but traditional computation is often expensive. This paper introduces CartNet, a novel graph neural network (GNN) for efficiently predicting crystal properties by encoding atomic geometry into Cartesian coordinates alongside the crystal temperature. CartNet integrates a neighbour equalization technique to emphasize covalent and contact interactions, and a Cholesky-based head to ensure valid ADP predictions. We also propose a rotational SO(3) data augmentation strategy during training to handle unseen orientations. An ADP dataset with over 200,000 experimental crystal structures from the Cambridge Structural Database (CSD) was curated to validate the approach. CartNet significantly reduces computational costs and outperforms existing methods in ADP prediction by 10.87%, while delivering a 34.77% improvement over theoretical approaches. We further evaluated CartNet on other datasets covering formation energy, band gap, total energy, energy above the convex hull, bulk moduli, and shear moduli, achieving 7.71% better results on the Jarvis Dataset and 13.16% on the Materials Project Dataset. These gains establish CartNet as a state-of-the-art solution for diverse crystal property predictions. Project website and online demo: https://www.ee.ub.edu/cartnet

We present CARTO, a novel approach for reconstructing multiple articulated objects from a single stereo RGB observation. We use implicit object-centric representations and learn a single geometry and articulation decoder for multiple object categories. Despite training on multiple categories, our decoder achieves a comparable reconstruction accuracy to methods that train bespoke decoders separately for each category. Combined with our stereo image encoder we infer the 3D shape, 6D pose, size, joint type, and the joint state of multiple unknown objects in a single forward pass. Our method achieves a 20.4% absolute improvement in mAP 3D IOU50 for novel instances when compared to a two-stage pipeline. Inference time is fast and can run on a NVIDIA TITAN XP GPU at 1 HZ for eight or less objects present. While only trained on simulated data, CARTO transfers to real-world object instances. Code and evaluation data is available at: http://carto.cs.uni-freiburg.de

In this paper, we study the ideas of composition and decomposition in the context of vector spaces, graphs and matroids. For vector spaces V_{AB}, treated as collection of row vectors, with specified column set Auplus B, we define V_{SP}lrarv V_{PQ}, Scap Q= emptyset, to be the collection of all vectors (f_S,f_Q) such that (f_S,f_P)in V_{SP}, (f_P,f_Q)in V_{PQ}. An analogous operation G_{SP}lrarg G_{PQ}equivd G_{PQ} can be defined in relation to graphs G_{SP}, G_{PQ}, on edge sets Suplus P, Puplus Q, respectively in terms of an overlapping subgraph G_P which gets deleted in the right side graph (see for instance the notion of k-sum oxley). For matroids we define the `linking' M_{SP}lrarm M_{PQ} equivd (M_{SP}vee M_{PQ})times (Suplus Q), denoting the contraction operation by 'times'. In each case, we examine how to minimize the size of the `overlap' set P, without affecting the right side entity. In the case of vector spaces, there is a polynomial time algorithm for achieving the minimum, which we present. Similar ideas work for graphs and for matroids under appropriate conditions. Next we consider the problem of decomposition. Here, in the case of vector spaces, the problem is to decompose V_{SQ} as V_{SP}lrarv V_{PQ}, with minimum size P. We give a polynomial time algorithm for this purpose. In the case of graphs and matroids we give a solution to this problem under certain restrictions.

We require decomposition methods for the ABCD-matrix formulation in rotationally symmetric paraxial geometric optics when designing a multi-component optical system from a given single paraxial specification (represented by an ABCD matrix) to optimize non-paraxial specifications (e.g., optical aberrations). In this study, we propose two kinds of three-matrix decomposition of ABCD matrices by focusing on the fact that the ABCD matrices have three real-number degrees of freedom. In addition, we formulate a transformation between the two kinds of decomposition for a single matrix, which can increase or decrease the number of refraction surfaces in the optical configuration while keeping the paraxial specifications fixed. This nature is useful for the optical design of multi-component systems with optimized non-paraxial characteristics.

Learning decompositions of expensive-to-evaluate black-box functions promises to scale Bayesian optimisation (BO) to high-dimensional problems. However, the success of these techniques depends on finding proper decompositions that accurately represent the black-box. While previous works learn those decompositions based on data, we investigate data-independent decomposition sampling rules in this paper. We find that data-driven learners of decompositions can be easily misled towards local decompositions that do not hold globally across the search space. Then, we formally show that a random tree-based decomposition sampler exhibits favourable theoretical guarantees that effectively trade off maximal information gain and functional mismatch between the actual black-box and its surrogate as provided by the decomposition. Those results motivate the development of the random decomposition upper-confidence bound algorithm (RDUCB) that is straightforward to implement - (almost) plug-and-play - and, surprisingly, yields significant empirical gains compared to the previous state-of-the-art on a comprehensive set of benchmarks. We also confirm the plug-and-play nature of our modelling component by integrating our method with HEBO, showing improved practical gains in the highest dimensional tasks from Bayesmark.

SO(3)-equivariant networks are the dominant models for machine learning interatomic potentials (MLIPs). The key operation of such networks is the Clebsch-Gordan (CG) tensor product, which is computationally expensive. To accelerate the computation, we develop tensor decomposition networks (TDNs) as a class of approximately equivariant networks in which CG tensor products are replaced by low-rank tensor decompositions, such as the CANDECOMP/PARAFAC (CP) decomposition. With the CP decomposition, we prove (i) a uniform bound on the induced error of SO(3)-equivariance, and (ii) the universality of approximating any equivariant bilinear map. To further reduce the number of parameters, we propose path-weight sharing that ties all multiplicity-space weights across the O(L^3) CG paths into a single shared parameter set without compromising equivariance, where L is the maximum angular degree. The resulting layer acts as a plug-and-play replacement for tensor products in existing networks, and the computational complexity of tensor products is reduced from O(L^6) to O(L^4). We evaluate TDNs on PubChemQCR, a newly curated molecular relaxation dataset containing 105 million DFT-calculated snapshots. We also use existing datasets, including OC20, and OC22. Results show that TDNs achieve competitive performance with dramatic speedup in computations. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS/tree/main/OpenMol/TDN{https://github.com/divelab/AIRS/}).

In this work we introduce a physically motivated method of performing disc/spheroid decomposition of simulated galaxies, which we apply to the Eagle sample. We make use of the HEALPix package to create Mollweide projections of the angular momentum map of each galaxy's stellar particles. A number of features arise on the angular momentum space which allows us to decompose galaxies and classify them into different morphological types. We assign stellar particles with angular separation of less/greater than 30 degrees from the densest grid cell on the angular momentum sphere to the disc/spheroid components, respectively. We analyse the spatial distribution for a subsample of galaxies and show that the surface density profiles of the disc and spheroid closely follow an exponential and a Sersic profile, respectively. In addition discs rotate faster, have smaller velocity dispersions, are younger and are more metal rich than spheroids. Thus our morphological classification reproduces the observed properties of such systems. Finally, we demonstrate that our method is able to identify a significant population of galaxies with counter-rotating discs and provide a more realistic classification of such systems compared to previous methods.

Reconstructing 3D human shape and pose from monocular images is challenging despite the promising results achieved by the most recent learning-based methods. The commonly occurred misalignment comes from the facts that the mapping from images to the model space is highly non-linear and the rotation-based pose representation of body models is prone to result in the drift of joint positions. In this work, we investigate learning 3D human shape and pose from dense correspondences of body parts and propose a Decompose-and-aggregate Network (DaNet) to address these issues. DaNet adopts the dense correspondence maps, which densely build a bridge between 2D pixels and 3D vertices, as intermediate representations to facilitate the learning of 2D-to-3D mapping. The prediction modules of DaNet are decomposed into one global stream and multiple local streams to enable global and fine-grained perceptions for the shape and pose predictions, respectively. Messages from local streams are further aggregated to enhance the robust prediction of the rotation-based poses, where a position-aided rotation feature refinement strategy is proposed to exploit spatial relationships between body joints. Moreover, a Part-based Dropout (PartDrop) strategy is introduced to drop out dense information from intermediate representations during training, encouraging the network to focus on more complementary body parts as well as neighboring position features. The efficacy of the proposed method is validated on both indoor and real-world datasets including Human3.6M, UP3D, COCO, and 3DPW, showing that our method could significantly improve the reconstruction performance in comparison with previous state-of-the-art methods. Our code is publicly available at https://hongwenzhang.github.io/dense2mesh .

A quadrilateral block decomposition of a planar domain is judged by whether it is complete, whether its elements are well shaped, and how many of its vertices are irregular. The last has a provable floor: the discrete Gauss-Bonnet identity enforces a lower bound on the total vertex irregularity of any all-quadrilateral mesh of a given domain purely based on its topology and corner angles. We train a reinforcement learning agent to build decompositions that reach this bound, which we call par. It acts directly on the mesh's half-edge data structure through local edits, with a policy network whose convolutions follow the mesh's own connectivity, so it applies unchanged to domains larger than any seen in training. The reward targets the floor directly, and it is sparse: random play reaches it on no domain with more than eight sides. We overcome this exploration barrier via behaviour cloning on optimal meshes that are trivial to construct, walked backward into demonstrations, before training it with PPO. On 96 held-out domains the agent produces an all-quadrilateral mesh on every one, a usable one on 95.7 on average, and a provably optimal one on 90; Gmsh's strongest configuration at the same element count completes 51, is usable on 38 and optimal on none, and even at three to fourteen times the elements never produces a more regular mesh. On 64 domains twice the training size the agent completes all, is usable on 62, and keeps a median excess over par below one against Gmsh's 39 at the same element count.

Transformer representations evolve through learned additive transformations that either preserve their current direction or redirect it. We study this evolution as a functional geometry, decomposing learned updates into parallel and perpendicular components. Across pretrained models, we find substantial parallel components beyond the residual identity path. We then apply the decomposition in two spaces: to attention and MLP updates relative to the hidden state, and to attention value aggregation relative to the current token's value. Targeted edits reveal a strongly space-dependent asymmetry: exclude-self value-space parallel manipulation is markedly more robust than residual-space and perpendicular counterparts, preserving the direct self message while scaling only the non-self aggregate. The same decomposition gives a component-resolved description of compression-induced update error: perpendicular error separates compression methods more clearly than parallel error. Extensive experiments further demonstrate that full-aggregate parallel suppression during from-scratch pretraining lowers validation-loss trajectories and improves downstream averages, with the value-space variant strongest. Together, these results connect representation geometry to editing robustness, compression diagnosis, and training-time intervention. Code is available in the https://github.com/Shwai-He/Transformer-Geometry{project repository}.

This work proposes a new formulation to the long-standing problem of convex decomposition through learning feature fields, enabling the first feed-forward model for open-world convex decomposition. Our method produces high-quality decompositions of 3D shapes into a union of convex bodies, which are essential to accelerate collision detection in physical simulation, amongst many other applications. The key insight is to adopt a feature learning approach and learn a continuous feature field that can later be clustered to yield a good convex decomposition via our self-supervised, purely-geometric objective derived from the classical definition of convexity. Our formulation can be used for single shape optimization, but more importantly, feature prediction unlocks scalable, self-supervised learning on large datasets resulting in the first learned open-world model for convex decomposition. Experiments show that our decompositions are higher-quality than alternatives and generalize across open-world objects as well as across representations to meshes, CAD models, and even Gaussian splats. https://research.nvidia.com/labs/sil/projects/learning-convex-decomp/

Intrinsic imaging or intrinsic image decomposition has traditionally been described as the problem of decomposing an image into two layers: a reflectance, the albedo invariant color of the material; and a shading, produced by the interaction between light and geometry. Deep learning techniques have been broadly applied in recent years to increase the accuracy of those separations. In this survey, we overview those results in context of well-known intrinsic image data sets and relevant metrics used in the literature, discussing their suitability to predict a desirable intrinsic image decomposition. Although the Lambertian assumption is still a foundational basis for many methods, we show that there is increasing awareness on the potential of more sophisticated physically-principled components of the image formation process, that is, optically accurate material models and geometry, and more complete inverse light transport estimations. We classify these methods in terms of the type of decomposition, considering the priors and models used, as well as the learning architecture and methodology driving the decomposition process. We also provide insights about future directions for research, given the recent advances in neural, inverse and differentiable rendering techniques.

Path-velocity decomposition is an intuitive yet powerful approach to address the complexity of kinodynamic motion planning. The difficult trajectory planning problem is solved in two separate, simpler, steps: first, find a path in the configuration space that satisfies the geometric constraints (path planning), and second, find a time-parameterization of that path satisfying the kinodynamic constraints. A fundamental requirement is that the path found in the first step should be time-parameterizable. Most existing works fulfill this requirement by enforcing quasi-static constraints in the path planning step, resulting in an important loss in completeness. We propose a method that enables path-velocity decomposition to discover truly dynamic motions, i.e. motions that are not quasi-statically executable. At the heart of the proposed method is a new algorithm -- Admissible Velocity Propagation -- which, given a path and an interval of reachable velocities at the beginning of that path, computes exactly and efficiently the interval of all the velocities the system can reach after traversing the path while respecting the system kinodynamic constraints. Combining this algorithm with usual sampling-based planners then gives rise to a family of new trajectory planners that can appropriately handle kinodynamic constraints while retaining the advantages associated with path-velocity decomposition. We demonstrate the efficiency of the proposed method on some difficult kinodynamic planning problems, where, in particular, quasi-static methods are guaranteed to fail.

Mechanistic interpretability aims to understand the internal mechanisms learned by neural networks. Despite recent progress toward this goal, it remains unclear how best to decompose neural network parameters into mechanistic components. We introduce Attribution-based Parameter Decomposition (APD), a method that directly decomposes a neural network's parameters into components that (i) are faithful to the parameters of the original network, (ii) require a minimal number of components to process any input, and (iii) are maximally simple. Our approach thus optimizes for a minimal length description of the network's mechanisms. We demonstrate APD's effectiveness by successfully identifying ground truth mechanisms in multiple toy experimental settings: Recovering features from superposition; separating compressed computations; and identifying cross-layer distributed representations. While challenges remain to scaling APD to non-toy models, our results suggest solutions to several open problems in mechanistic interpretability, including identifying minimal circuits in superposition, offering a conceptual foundation for 'features', and providing an architecture-agnostic framework for neural network decomposition.

With the advent of Neural Radiance Fields (NeRF), neural networks can now render novel views of a 3D scene with quality that fools the human eye. Yet, generating these images is very computationally intensive, limiting their applicability in practical scenarios. In this paper, we propose a technique based on spatial decomposition capable of mitigating this issue. Our key observation is that there are diminishing returns in employing larger (deeper and/or wider) networks. Hence, we propose to spatially decompose a scene and dedicate smaller networks for each decomposed part. When working together, these networks can render the whole scene. This allows us near-constant inference time regardless of the number of decomposed parts. Moreover, we show that a Voronoi spatial decomposition is preferable for this purpose, as it is provably compatible with the Painter's Algorithm for efficient and GPU-friendly rendering. Our experiments show that for real-world scenes, our method provides up to 3x more efficient inference than NeRF (with the same rendering quality), or an improvement of up to 1.0~dB in PSNR (for the same inference cost).

We present an algorithm for decomposing low rank tensors of any symmetry type, from fully asymmetric to fully symmetric. It generalizes the recent subspace power method from symmetric tensors to all tensors. The algorithm transforms an input tensor into a tensor with orthonormal slices. We show that for tensors with orthonormal slices and low rank, the summands of their decomposition are in one-to-one correspondence with the partially symmetric singular vector tuples (pSVTs) with singular value one. We use this to show correctness of the algorithm. We introduce a shifted power method for computing pSVTs and establish its global convergence. Numerical experiments demonstrate that our decomposition algorithm achieves higher accuracy and faster runtime than existing methods.

A key step in reverse engineering neural networks is to decompose them into simpler parts that can be studied in relative isolation. Linear parameter decomposition -- a framework that has been proposed to resolve several issues with current decomposition methods -- decomposes neural network parameters into a sum of sparsely used vectors in parameter space. However, the current main method in this framework, Attribution-based Parameter Decomposition (APD), is impractical on account of its computational cost and sensitivity to hyperparameters. In this work, we introduce Stochastic Parameter Decomposition (SPD), a method that is more scalable and robust to hyperparameters than APD, which we demonstrate by decomposing models that are slightly larger and more complex than was possible to decompose with APD. We also show that SPD avoids other issues, such as shrinkage of the learned parameters, and better identifies ground truth mechanisms in toy models. By bridging causal mediation analysis and network decomposition methods, this demonstration opens up new research possibilities in mechanistic interpretability by removing barriers to scaling linear parameter decomposition methods to larger models. We release a library for running SPD and reproducing our experiments at https://github.com/goodfire-ai/spd.

A Lie group is an old mathematical abstract object dating back to the XIX century, when mathematician Sophus Lie laid the foundations of the theory of continuous transformation groups. As it often happens, its usage has spread over diverse areas of science and technology many years later. In robotics, we are recently experiencing an important trend in its usage, at least in the fields of estimation, and particularly in motion estimation for navigation. Yet for a vast majority of roboticians, Lie groups are highly abstract constructions and therefore difficult to understand and to use. This may be due to the fact that most of the literature on Lie theory is written by and for mathematicians and physicists, who might be more used than us to the deep abstractions this theory deals with. In estimation for robotics it is often not necessary to exploit the full capacity of the theory, and therefore an effort of selection of materials is required. In this paper, we will walk through the most basic principles of the Lie theory, with the aim of conveying clear and useful ideas, and leave a significant corpus of the Lie theory behind. Even with this mutilation, the material included here has proven to be extremely useful in modern estimation algorithms for robotics, especially in the fields of SLAM, visual odometry, and the like. Alongside this micro Lie theory, we provide a chapter with a few application examples, and a vast reference of formulas for the major Lie groups used in robotics, including most jacobian matrices and the way to easily manipulate them. We also present a new C++ template-only library implementing all the functionality described here.

The ability to decompose scenes in terms of abstract building blocks is crucial for general intelligence. Where those basic building blocks share meaningful properties, interactions and other regularities across scenes, such decompositions can simplify reasoning and facilitate imagination of novel scenarios. In particular, representing perceptual observations in terms of entities should improve data efficiency and transfer performance on a wide range of tasks. Thus we need models capable of discovering useful decompositions of scenes by identifying units with such regularities and representing them in a common format. To address this problem, we have developed the Multi-Object Network (MONet). In this model, a VAE is trained end-to-end together with a recurrent attention network -- in a purely unsupervised manner -- to provide attention masks around, and reconstructions of, regions of images. We show that this model is capable of learning to decompose and represent challenging 3D scenes into semantically meaningful components, such as objects and background elements.

Recent advances in 3D X-ray Computed Tomographic (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Voxel-based description (up to hundreds millions voxels) of the pore space can be extracted, from grey level 3D CT scanner images, by means of simple image processing tools. Classical methods for numerical simulation of biological dynamics using mesh of voxels, such as Lattice Boltzmann Model (LBM), are too much time consuming. Thus, the use of more compact and reliable geometrical representations of pore space can drastically decrease the computational cost of the simulations. Several recent works propose basic analytic volume primitives (e.g. spheres, generalized cylinders, ellipsoids) to define a piece-wise approximation of pore space for numerical simulation of draining, diffusion and microbial decomposition. Such approaches work well but the drawback is that it generates approximation errors. In the present work, we study another alternative where pore space is described by means of geometrically relevant connected subsets of voxels (regions) computed from the curvilinear skeleton. Indeed, many works use the curvilinear skeleton (3D medial axis) for analyzing and partitioning 3D shapes within various domains (medicine, material sciences, petroleum engineering, etc.) but only a few ones in soil sciences. Within the context of soil sciences, most studies dealing with 3D medial axis focus on the determination of pore throats. Here, we segment pore space using curvilinear skeleton in order to achieve numerical simulation of microbial decomposition (including diffusion processes). We validate simulation outputs by comparison with other methods using different pore space geometrical representations (balls, voxels).

Large Language Models (LLMs) prompted to generate chain-of-thought (CoT) exhibit impressive reasoning capabilities. Recent attempts at prompt decomposition toward solving complex, multi-step reasoning problems depend on the ability of the LLM to simultaneously decompose and solve the problem. A significant disadvantage is that foundational LLMs are typically not available for fine-tuning, making adaptation computationally prohibitive. We believe (and demonstrate) that problem decomposition and solution generation are distinct capabilites, better addressed in separate modules, than by one monolithic LLM. We introduce DaSLaM, which uses a decomposition generator to decompose complex problems into subproblems that require fewer reasoning steps. These subproblems are answered by a solver. We use a relatively small (13B parameters) LM as the decomposition generator, which we train using policy gradient optimization to interact with a solver LM (regarded as black-box) and guide it through subproblems, thereby rendering our method solver-agnostic. Evaluation on multiple different reasoning datasets reveal that with our method, a 175 billion parameter LM (text-davinci-003) can produce competitive or even better performance, compared to its orders-of-magnitude larger successor, GPT-4. Additionally, we show that DaSLaM is not limited by the solver's capabilities as a function of scale; e.g., solver LMs with diverse sizes give significant performance improvement with our solver-agnostic decomposition technique. Exhaustive ablation studies evince the superiority of our modular finetuning technique over exorbitantly large decomposer LLMs, based on prompting alone.

Self-supervised learning (SSL) pipelines differ in many design choices such as the architecture, augmentations, or pretraining data. Yet SSL is typically evaluated using a single metric: linear probing on ImageNet. This does not provide much insight into why or when a model is better, now how to improve it. To address this, we propose an SSL risk decomposition, which generalizes the classical supervised approximation-estimation decomposition by considering errors arising from the representation learning step. Our decomposition consists of four error components: approximation, representation usability, probe generalization, and encoder generalization. We provide efficient estimators for each component and use them to analyze the effect of 30 design choices on 169 SSL vision models evaluated on ImageNet. Our analysis gives valuable insights for designing and using SSL models. For example, it highlights the main sources of error and shows how to improve SSL in specific settings (full- vs few-shot) by trading off error components. All results and pretrained models are at https://github.com/YannDubs/SSL-Risk-Decomposition.

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of O left(varepsilon^{-(3α-2){(α-1)}} right) for finding an varepsilon-stationary point, where αin(1,2] denotes the heavy-tail index. For the inexact-polar setting with σ_1=0, we also provide guarantees that do not require prior knowledge of α. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

This paper makes a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which results in the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. A second important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

Determining the stage of decomposition (SOD) is crucial for estimating the postmortem interval and identifying human remains. Currently, labor-intensive manual scoring methods are used for this purpose, but they are subjective and do not scale for the emerging large-scale archival collections of human decomposition photos. This study explores the feasibility of automating two common human decomposition scoring methods proposed by Megyesi and Gelderman using artificial intelligence (AI). We evaluated two popular deep learning models, Inception V3 and Xception, by training them on a large dataset of human decomposition images to classify the SOD for different anatomical regions, including the head, torso, and limbs. Additionally, an interrater study was conducted to assess the reliability of the AI models compared to human forensic examiners for SOD identification. The Xception model achieved the best classification performance, with macro-averaged F1 scores of .878, .881, and .702 for the head, torso, and limbs when predicting Megyesi's SODs, and .872, .875, and .76 for the head, torso, and limbs when predicting Gelderman's SODs. The interrater study results supported AI's ability to determine the SOD at a reliability level comparable to a human expert. This work demonstrates the potential of AI models trained on a large dataset of human decomposition images to automate SOD identification.

Recently, triple decomposition has attracted increasing attention for decomposing third-order tensors into three factor tensors. However, this approach is limited to third-order tensors and enforces uniformity in the lower dimensions across all factor tensors, which restricts its flexibility and applicability. To address these issues, we propose the Multiple decomposition, a novel framework that generalizes triple decomposition to arbitrary order tensors and allows the short dimensions of the factor tensors to differ. We establish its connections with other classical tensor decompositions. Furthermore, implicit neural representation (INR) is employed to continuously represent the factor tensors in Multiple decomposition, enabling the method to generalize to non-grid data. We refer to this INR-based Multiple decomposition as Implicit Multiple Tensor Decomposition (IMTD). Then, the Proximal Alternating Least Squares (PALS) algorithm is utilized to solve the IMTD-based tensor reconstruction models. Since the objective function in IMTD-based models often lacks the Kurdyka-Lojasiewicz (KL) property, we establish a KL-free convergence analysis for the algorithm. Finally, extensive numerical experiments further validate the effectiveness of the proposed method.

Explicit decomposition modeling, which involves breaking down complex tasks into more straightforward and often more interpretable sub-tasks, has long been a central theme in developing robust and interpretable NLU systems. However, despite the many datasets and resources built as part of this effort, the majority have small-scale annotations and limited scope, which is insufficient to solve general decomposition tasks. In this paper, we look at large-scale intermediate pre-training of decomposition-based transformers using distant supervision from comparable texts, particularly large-scale parallel news. We show that with such intermediate pre-training, developing robust decomposition-based models for a diverse range of tasks becomes more feasible. For example, on semantic parsing, our model, DecompT5, improves 20% to 30% on two datasets, Overnight and TORQUE, over the baseline language model. We further use DecompT5 to build a novel decomposition-based QA system named DecompEntail, improving over state-of-the-art models, including GPT-3, on both HotpotQA and StrategyQA by 8% and 4%, respectively.

Representation learning and exploration are among the key challenges for any deep reinforcement learning agent. In this work, we provide a singular value decomposition based method that can be used to obtain representations that preserve the underlying transition structure in the domain. Perhaps interestingly, we show that these representations also capture the relative frequency of state visitations, thereby providing an estimate for pseudo-counts for free. To scale this decomposition method to large-scale domains, we provide an algorithm that never requires building the transition matrix, can make use of deep networks, and also permits mini-batch training. Further, we draw inspiration from predictive state representations and extend our decomposition method to partially observable environments. With experiments on multi-task settings with partially observable domains, we show that the proposed method can not only learn useful representation on DM-Lab-30 environments (that have inputs involving language instructions, pixel images, and rewards, among others) but it can also be effective at hard exploration tasks in DM-Hard-8 environments.

Deep Neural Networks (DNNs) have been a large driver and enabler for AI breakthroughs in recent years. These models have been getting larger in their attempt to become more accurate and tackle new upcoming use-cases, including AR/VR and intelligent assistants. However, the training process of such large models is a costly and time-consuming process, which typically yields a single model to fit all targets. To mitigate this, various techniques have been proposed in the literature, including pruning, sparsification or quantization of the model weights and updates. While able to achieve high compression rates, they often incur computational overheads or accuracy penalties. Alternatively, factorization methods have been leveraged to incorporate low-rank compression in the training process. Similarly, such techniques (e.g.,~SVD) frequently rely on the computationally expensive decomposition of layers and are potentially sub-optimal for non-linear models, such as DNNs. In this work, we take a further step in designing efficient low-rank models and propose Maestro, a framework for trainable low-rank layers. Instead of regularly applying a priori decompositions such as SVD, the low-rank structure is built into the training process through a generalized variant of Ordered Dropout. This method imposes an importance ordering via sampling on the decomposed DNN structure. Our theoretical analysis demonstrates that our method recovers the SVD decomposition of linear mapping on uniformly distributed data and PCA for linear autoencoders. We further apply our technique on DNNs and empirically illustrate that Maestro enables the extraction of lower footprint models that preserve model performance while allowing for graceful accuracy-latency tradeoff for the deployment to devices of different capabilities.

Recent advances in video diffusion models have demonstrated strong potential for generating robotic decision-making data, with trajectory conditions further enabling fine-grained control. However, existing trajectory-based methods primarily focus on individual object motion and struggle to capture multi-object interaction crucial in complex robotic manipulation. This limitation arises from multi-feature entanglement in overlapping regions, which leads to degraded visual fidelity. To address this, we present RoboMaster, a novel framework that models inter-object dynamics through a collaborative trajectory formulation. Unlike prior methods that decompose objects, our core is to decompose the interaction process into three sub-stages: pre-interaction, interaction, and post-interaction. Each stage is modeled using the feature of the dominant object, specifically the robotic arm in the pre- and post-interaction phases and the manipulated object during interaction, thereby mitigating the drawback of multi-object feature fusion present during interaction in prior work. To further ensure subject semantic consistency throughout the video, we incorporate appearance- and shape-aware latent representations for objects. Extensive experiments on the challenging Bridge V2 dataset, as well as in-the-wild evaluation, demonstrate that our method outperforms existing approaches, establishing new state-of-the-art performance in trajectory-controlled video generation for robotic manipulation.

Low Rank Decomposition of matrix - splitting a large matrix into a product of two smaller matrix offers a means for compression that reduces the parameters of a model without sparsification, and hence delivering more speedup on modern hardware. Moreover, unlike quantization, the compressed linear layers remain fully differentiable and all the parameters trainable, while being able to leverage the existing highly efficient kernels over floating point matrices. We study the potential to compress Large Language Models (LLMs) for monolingual Code generation via Low Rank Decomposition (LoRD) and observe that ranks for the linear layers in these models can be reduced by upto 39.58% with less than 1% increase in perplexity. We then use Low Rank Decomposition (LoRD) to compress StarCoder 16B to 13.2B parameter with no drop and to 12.3B with minimal drop in HumanEval Pass@1 score, in less than 10 minutes on a single A100. The compressed models speeds up inference by up to 22.35% with just a single line of change in code over huggingface's implementation with pytorch backend. Low Rank Decomposition (LoRD) models remain compatible with state of the art near-lossless quantization method such as SpQR, which allows leveraging further compression gains of quantization. Lastly, QLoRA over Low Rank Decomposition (LoRD) model further reduces memory requirements by as much as 21.2% over vanilla QLoRA while offering similar gains from parameter efficient fine tuning. Our work shows Low Rank Decomposition (LoRD) as a promising new paradigm for LLM compression.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

Cooperative multi-agent reinforcement learning (MARL) commonly adopts centralized training with decentralized execution (CTDE), where centralized critics leverage global information to guide decentralized actors. However, centralized-decentralized mismatch (CDM) arises when the suboptimal behavior of one agent degrades others' learning. Prior approaches mitigate CDM through value decomposition, but linear decompositions allow per-agent gradients at the cost of limited expressiveness, while nonlinear decompositions improve representation but require centralized gradients, reintroducing CDM. To overcome this trade-off, we propose the multi-agent cross-entropy method (MCEM), combined with monotonic nonlinear critic decomposition (NCD). MCEM updates policies by increasing the probability of high-value joint actions, thereby excluding suboptimal behaviors. For sample efficiency, we extend off-policy learning with a modified k-step return and Retrace. Analysis and experiments demonstrate that MCEM outperforms state-of-the-art methods across both continuous and discrete action benchmarks.

We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a d-dimensional lattice with periodic boundary conditions and n = L^d sites, the Fisher manifold has m = d cdot n dimensions (one per bond), and we find |R(J_c)| sim n^{d_R} with d_R = (dν+ 2η)/(dν+ η), where ν and η are the correlation-length and anomalous-dimension critical exponents. For 2D Ising (ν= 1, η= 1/4), this predicts d_R = 10/9, confirmed by exact transfer-matrix computations (L = 6--9: d_R = 1.1115 pm 0.0002) and multi-seed MCMC through L = 24. For 3D Ising (ν= 0.630, η= 0.0363), the prediction d_R = 1.019 is consistent with MCMC on L^3 tori up to L = 10 (power-law fit: d_R = 1.040). For 2D Potts q = 3 (predicted 33/29 approx 1.138), FFT-MCMC through L = 40 shows d_eff oscillating non-monotonically around sim 1.20, consistent with O(1/(ln L)^2) logarithmic corrections. For q = 4 (predicted 22/19), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity R_3 = -R_1/2, R_4 = -R_2/2 holds to 5--6 digits for all models. This exponent is distinct from Ruppeiner thermodynamic curvature and reflects the collective geometry of the growing Fisher manifold. We provide falsification criteria and predictions for additional universality classes.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

In this article, we focus on decomposing latent representations in generative adversarial networks or learned feature representations in deep autoencoders into semantically controllable factors in a semisupervised manner, without modifying the original trained models. Particularly, we propose factors' decomposer-entangler network (FDEN) that learns to decompose a latent representation into mutually independent factors. Given a latent representation, the proposed framework draws a set of interpretable factors, each aligned to independent factors of variations by minimizing their total correlation in an information-theoretic means. As a plug-in method, we have applied our proposed FDEN to the existing networks of adversarially learned inference and pioneer network and performed computer vision tasks of image-to-image translation in semantic ways, e.g., changing styles, while keeping the identity of a subject, and object classification in a few-shot learning scheme. We have also validated the effectiveness of the proposed method with various ablation studies in the qualitative, quantitative, and statistical examination.

Feature engineering is required to obtain better results for time series forecasting, and decomposition is a crucial one. One decomposition approach often cannot be used for numerous forecasting tasks since the standard time series decomposition lacks flexibility and robustness. Traditional feature selection relies heavily on preexisting domain knowledge, has no generic methodology, and requires a lot of labor. However, most time series prediction models based on deep learning typically suffer from interpretability issue, so the "black box" results lead to a lack of confidence. To deal with the above issues forms the motivation of the thesis. In the paper we propose TSDFNet as a neural network with self-decomposition mechanism and an attentive feature fusion mechanism, It abandons feature engineering as a preprocessing convention and creatively integrates it as an internal module with the deep model. The self-decomposition mechanism empowers TSDFNet with extensible and adaptive decomposition capabilities for any time series, users can choose their own basis functions to decompose the sequence into temporal and generalized spatial dimensions. Attentive feature fusion mechanism has the ability to capture the importance of external variables and the causality with target variables. It can automatically suppress the unimportant features while enhancing the effective ones, so that users do not have to struggle with feature selection. Moreover, TSDFNet is easy to look into the "black box" of the deep neural network by feature visualization and analyze the prediction results. We demonstrate performance improvements over existing widely accepted models on more than a dozen datasets, and three experiments showcase the interpretability of TSDFNet.

Decompositional reconstruction of 3D scenes, with complete shapes and detailed texture of all objects within, is intriguing for downstream applications but remains challenging, particularly with sparse views as input. Recent approaches incorporate semantic or geometric regularization to address this issue, but they suffer significant degradation in underconstrained areas and fail to recover occluded regions. We argue that the key to solving this problem lies in supplementing missing information for these areas. To this end, we propose DP-Recon, which employs diffusion priors in the form of Score Distillation Sampling (SDS) to optimize the neural representation of each individual object under novel views. This provides additional information for the underconstrained areas, but directly incorporating diffusion prior raises potential conflicts between the reconstruction and generative guidance. Therefore, we further introduce a visibility-guided approach to dynamically adjust the per-pixel SDS loss weights. Together these components enhance both geometry and appearance recovery while remaining faithful to input images. Extensive experiments across Replica and ScanNet++ demonstrate that our method significantly outperforms SOTA methods. Notably, it achieves better object reconstruction under 10 views than the baselines under 100 views. Our method enables seamless text-based editing for geometry and appearance through SDS optimization and produces decomposed object meshes with detailed UV maps that support photorealistic Visual effects (VFX) editing. The project page is available at https://dp-recon.github.io/.

Low-rank decomposition has emerged as a vital tool for enhancing parameter efficiency in neural network architectures, gaining traction across diverse applications in machine learning. These techniques significantly lower the number of parameters, striking a balance between compactness and performance. However, a common challenge has been the compromise between parameter efficiency and the accuracy of the model, where reduced parameters often lead to diminished accuracy compared to their full-rank counterparts. In this work, we propose a novel theoretical framework that integrates a sinusoidal function within the low-rank decomposition process. This approach not only preserves the benefits of the parameter efficiency characteristic of low-rank methods but also increases the decomposition's rank, thereby enhancing model performance. Our method proves to be a plug in enhancement for existing low-rank models, as evidenced by its successful application in Vision Transformers (ViT), Large Language Models (LLMs), Neural Radiance Fields (NeRF) and 3D shape modelling.

Every Cartesian three-dimensional PDE solver hides a structural secret that production CFD codes have used for half a century and that graduate-level textbooks rarely state plainly. The derivative matrices, the compact Padé line solves, the Galerkin mass inversions, the alternating-direction-implicit substeps, and even the fast Poisson and Helmholtz diagonalization transforms all factor along the coordinate axes and collapse into repeated one-dimensional banded kernels executed along the grid lines. The three-dimensional operator exists only on paper; it is never assembled, factored, or stored. This paper is the manual for that collapse. We derive the Kronecker-product algebra that makes it exact, carry it cleanly through central differences, compact schemes, tensor-product Galerkin, B-spline and isogeometric methods, collocation, ADI time stepping, and direct Poisson and Helmholtz solves, and bring into the open the three production tricks that turn the reduction into hardware-conscious floating-point throughput on real machines: the multi-right-hand-side reshape that exposes a sweep as one batched line kernel (a dense BLAS-3 GEMM when the line factor is dense or element-local, a banded or stencil kernel when it is not), the sum factorization that rescues high-order Galerkin from the O(p^{2d}) quadrature trap, and the pencil decomposition that keeps every direction contiguous across an MPI cluster. For fixed stencil width or fixed polynomial degree, the compute cost stays O(N) in the total number of unknowns N = N_x N_y N_z; the operator storage drops to O(N_x + N_y + N_z) up to bandwidth constants; direct separable Poisson and Helmholtz solvers add the expected transform cost; the line kernels are embarrassingly parallel. These facts are familiar to practitioners but rarely assembled in one place; this paper collects them and shows how to use them.

Disentangling visual layers in real-world images is a persistent challenge in vision and graphics, as such layers often involve non-linear and globally coupled interactions, including shading, reflection, and perspective distortion. In this work, we present an in-context image decomposition framework that leverages large diffusion foundation models for layered separation. We focus on the challenging case of logo-object decomposition, where the goal is to disentangle a logo from the surface on which it appears while faithfully preserving both layers. Our method fine-tunes a pretrained diffusion model via lightweight LoRA adaptation and introduces a cycle-consistent tuning strategy that jointly trains decomposition and composition models, enforcing reconstruction consistency between decomposed and recomposed images. This bidirectional supervision substantially enhances robustness in cases where the layers exhibit complex interactions. Furthermore, we introduce a progressive self-improving process, which iteratively augments the training set with high-quality model-generated examples to refine performance. Extensive experiments demonstrate that our approach achieves accurate and coherent decompositions and also generalizes effectively across other decomposition types, suggesting its potential as a unified framework for layered image decomposition.

We construct S-linear semiorthogonal decompositions of derived categories of smooth Fano threefold fibrations X/S with relative Picard rank 1 and rational geometric fibers and discuss how the structure of components of these decompositions is related to rationality properties of X/S.

As large language models (LLMs) perform more difficult tasks, it becomes harder to verify the correctness and safety of their behavior. One approach to help with this issue is to prompt LLMs to externalize their reasoning, e.g., by having them generate step-by-step reasoning as they answer a question (Chain-of-Thought; CoT). The reasoning may enable us to check the process that models use to perform tasks. However, this approach relies on the stated reasoning faithfully reflecting the model's actual reasoning, which is not always the case. To improve over the faithfulness of CoT reasoning, we have models generate reasoning by decomposing questions into subquestions. Decomposition-based methods achieve strong performance on question-answering tasks, sometimes approaching that of CoT while improving the faithfulness of the model's stated reasoning on several recently-proposed metrics. By forcing the model to answer simpler subquestions in separate contexts, we greatly increase the faithfulness of model-generated reasoning over CoT, while still achieving some of the performance gains of CoT. Our results show it is possible to improve the faithfulness of model-generated reasoning; continued improvements may lead to reasoning that enables us to verify the correctness and safety of LLM behavior.

Sparse Autoencoders (SAEs) are widely used for mechanistic interpretability in large language models, yet their formulation assigns each latent feature a single decoder direction, implicitly assuming features to be one-dimensional. We show that this assumption mismatches with the multi-dimensional structure of model features, provably inducing feature splitting through two distinct mechanisms. Geometrically, reconstructing a feature of intrinsic dimension d_i ge 2 to error varepsilon with single-direction decoders forces a number of atoms that is exponential in d_i. From an end-to-end optimization perspective, this splitting is not merely possible but actively preferred. We prove that there exists a continuous path from the true d_i-dimensional basis to a strictly lower risk of the ell_1-regularized SAE objective, whose descent directions drive any trained dictionary into that exponential regime. A single coherent feature is therefore fragmented across many near-collinear latents, producing spurious multiplicity and obscuring the intrinsic geometry. Motivated by this, we introduce Subspace-Aware Sparse Autoencoders (SASA), which replace single-vector decoders with learned decoder subspaces, enforce block sparsity via Top-s group gating, and adapt each group's effective rank with a nuclear-norm regularizer. We then show that once the block size satisfies r ge d_i, a single group not only can represent the entire feature slice but is the global minimizer of the SASA objective. This consolidation yields a sample complexity polynomial in d_i rather than exponential -- a decisive advantage given that every training activation costs an LLM forward pass. Empirically, on GPT-2 and Mistral-7B, SASA reduces feature splitting and absorption, improves monosemanticity and interpretability, and matches or exceeds standard SAEs while training on roughly half the token budget.

Continuous physical fields represent a large fraction of data under scientific investigation. Their multiscale structures are central to discovery, yet useful coordinates are not known in advance. Standard self-supervised methods define context and targets in fixed image coordinates, posing a predictive task misaligned with fields organized across a continuous scale hierarchy. We introduce ScaleAware-JEPA, a framework that constructs dense, label-free latent coordinates for continuous scalar fields. Constrained Diffusion Decomposition (CDD) separates each field into pixel-registered scale components and provides the scale coordinates that define the masking geometry. The resulting JEPA objective predicts hidden structure with a context footprint tied to the diffusion scale of each component rather than to an arbitrary patch size. Across MHD turbulence, interstellar molecular gas and urban nighttime-light structure, the learned geometry maps back to coherent morphology, forming dense structural atlases without labels or predefined segmentation rules. By tying latent prediction to the scale hierarchy of a field, ScaleAware-JEPA constructs latent coordinates through which complex physical patterns can be inspected before their relevant structures have been prescribed. Code is available at https://github.com/gxli/SA-JEPA.

A common goal of mechanistic interpretability is to decompose the activations of neural networks into features: interpretable properties of the input computed by the model. Sparse autoencoders (SAEs) are a popular method for finding these features in LLMs, and it has been postulated that they can be used to find a canonical set of units: a unique and complete list of atomic features. We cast doubt on this belief using two novel techniques: SAE stitching to show they are incomplete, and meta-SAEs to show they are not atomic. SAE stitching involves inserting or swapping latents from a larger SAE into a smaller one. Latents from the larger SAE can be divided into two categories: novel latents, which improve performance when added to the smaller SAE, indicating they capture novel information, and reconstruction latents, which can replace corresponding latents in the smaller SAE that have similar behavior. The existence of novel features indicates incompleteness of smaller SAEs. Using meta-SAEs -- SAEs trained on the decoder matrix of another SAE -- we find that latents in SAEs often decompose into combinations of latents from a smaller SAE, showing that larger SAE latents are not atomic. The resulting decompositions are often interpretable; e.g. a latent representing ``Einstein'' decomposes into ``scientist'', ``Germany'', and ``famous person''. Even if SAEs do not find canonical units of analysis, they may still be useful tools. We suggest that future research should either pursue different approaches for identifying such units, or pragmatically choose the SAE size suited to their task. We provide an interactive dashboard to explore meta-SAEs: https://metasaes.streamlit.app/

Latent Diffusion Models (LDMs) achieve high-fidelity synthesis but suffer from latent space brittleness, causing discontinuous semantic jumps during editing. We introduce a Riemannian framework to diagnose this instability by analyzing the generative Jacobian, decomposing geometry into Local Scaling (capacity) and Local Complexity (curvature). Our study uncovers a ``Geometric Decoupling": while curvature in normal generation functionally encodes image detail, OOD generation exhibits a functional decoupling where extreme curvature is wasted on unstable semantic boundaries rather than perceptible details. This geometric misallocation identifies ``Geometric Hotspots" as the structural root of instability, providing a robust intrinsic metric for diagnosing generative reliability.

Given a complex of groups G(Y) = (G_sigma, psi_a, g_{a,b}) where all G_sigma are relatively hyperbolic, the psi_a are inclusions of full relatively quasiconvex subgroups, and the universal cover X is CAT(0) and delta--hyperbolic, we show pi_1(G(Y)) is relatively hyperbolic. The proof extends the work of Dahmani and Martin by constructing a model for the Bowditch boundary of pi_1(G(Y)). We prove the model is a compact metrizable space on which G acts as a geometrically finite convergence group, and a theorem of Yaman then implies the result. More generally, this model shows how any suitable action of a relatively hyperbolic group on a simply connected cell complex encodes a decomposition of the Bowditch boundary into the boundary of the cell complex and the boundaries of cell stabilizers. We hope this decomposition will be helpful in answering topological questions about Bowditch boundaries.

Decomposing a scene into its shape, reflectance and illumination is a fundamental problem in computer vision and graphics. Neural approaches such as NeRF have achieved remarkable success in view synthesis, but do not explicitly perform decomposition and instead operate exclusively on radiance (the product of reflectance and illumination). Extensions to NeRF, such as NeRD, can perform decomposition but struggle to accurately recover detailed illumination, thereby significantly limiting realism. We propose a novel reflectance decomposition network that can estimate shape, BRDF, and per-image illumination given a set of object images captured under varying illumination. Our key technique is a novel illumination integration network called Neural-PIL that replaces a costly illumination integral operation in the rendering with a simple network query. In addition, we also learn deep low-dimensional priors on BRDF and illumination representations using novel smooth manifold auto-encoders. Our decompositions can result in considerably better BRDF and light estimates enabling more accurate novel view-synthesis and relighting compared to prior art. Project page: https://markboss.me/publication/2021-neural-pil/

3D creation has always been a unique human strength, driven by our ability to deconstruct and reassemble objects using our eyes, mind and hand. However, current 3D design tools struggle to replicate this natural process, requiring considerable artistic expertise and manual labor. This paper introduces BANG, a novel generative approach that bridges 3D generation and reasoning, allowing for intuitive and flexible part-level decomposition of 3D objects. At the heart of BANG is "Generative Exploded Dynamics", which creates a smooth sequence of exploded states for an input geometry, progressively separating parts while preserving their geometric and semantic coherence. BANG utilizes a pre-trained large-scale latent diffusion model, fine-tuned for exploded dynamics with a lightweight exploded view adapter, allowing precise control over the decomposition process. It also incorporates a temporal attention module to ensure smooth transitions and consistency across time. BANG enhances control with spatial prompts, such as bounding boxes and surface regions, enabling users to specify which parts to decompose and how. This interaction can be extended with multimodal models like GPT-4, enabling 2D-to-3D manipulations for more intuitive and creative workflows. The capabilities of BANG extend to generating detailed part-level geometry, associating parts with functional descriptions, and facilitating component-aware 3D creation and manufacturing workflows. Additionally, BANG offers applications in 3D printing, where separable parts are generated for easy printing and reassembly. In essence, BANG enables seamless transformation from imaginative concepts to detailed 3D assets, offering a new perspective on creation that resonates with human intuition.

This work focuses on the 3D reconstruction of non-rigid objects based on monocular RGB video sequences. Concretely, we aim at building high-fidelity models for generic object categories and casually captured scenes. To this end, we do not assume known root poses of objects, and do not utilize category-specific templates or dense pose priors. The key idea of our method, Root Pose Decomposition (RPD), is to maintain a per-frame root pose transformation, meanwhile building a dense field with local transformations to rectify the root pose. The optimization of local transformations is performed by point registration to the canonical space. We also adapt RPD to multi-object scenarios with object occlusions and individual differences. As a result, RPD allows non-rigid 3D reconstruction for complicated scenarios containing objects with large deformations, complex motion patterns, occlusions, and scale diversities of different individuals. Such a pipeline potentially scales to diverse sets of objects in the wild. We experimentally show that RPD surpasses state-of-the-art methods on the challenging DAVIS, OVIS, and AMA datasets.

In the context of C^* dynamical systems, we consider a locally compact group G acting by ^*-automorphisms on a C^* algebra U of observables, and assume a state of U that satisfies the clustering property with respect to a net of group elements of G. That is, the two-point connected correlation function vanishes in the limit on the net, when one observable is translated under the group action. Then we show that all higher order connected correlation functions (Ursell functions, or classical cumulants) and all free correlation functions (free cumulants, from free probability) vanish at the same rate in that limit. Additionally, we show that mean clustering, also called ergodicity, extends to higher order correlations. We then apply those results to equilibrium states of quantum spin lattice models. Under certain assumptions on the range of the interaction, high temperature Gibbs states are known to be exponentially clustering w.r.t. space translations. Combined with the Lieb-Robinson bound, one obtains exponential clustering for space-time translations outside the Lieb-Robinson light-cone. Therefore, by our present results, all the higher order connected and free correlation functions will vanish exponentially under space-time translations outside the Lieb-Robinson light cone, in high temperature Gibbs states. Another consequence is that their long-time averaging over a space-time ray vanishes for almost every ray velocity.

We present StdGEN, an innovative pipeline for generating semantically decomposed high-quality 3D characters from single images, enabling broad applications in virtual reality, gaming, and filmmaking, etc. Unlike previous methods which struggle with limited decomposability, unsatisfactory quality, and long optimization times, StdGEN features decomposability, effectiveness and efficiency; i.e., it generates intricately detailed 3D characters with separated semantic components such as the body, clothes, and hair, in three minutes. At the core of StdGEN is our proposed Semantic-aware Large Reconstruction Model (S-LRM), a transformer-based generalizable model that jointly reconstructs geometry, color and semantics from multi-view images in a feed-forward manner. A differentiable multi-layer semantic surface extraction scheme is introduced to acquire meshes from hybrid implicit fields reconstructed by our S-LRM. Additionally, a specialized efficient multi-view diffusion model and an iterative multi-layer surface refinement module are integrated into the pipeline to facilitate high-quality, decomposable 3D character generation. Extensive experiments demonstrate our state-of-the-art performance in 3D anime character generation, surpassing existing baselines by a significant margin in geometry, texture and decomposability. StdGEN offers ready-to-use semantic-decomposed 3D characters and enables flexible customization for a wide range of applications. Project page: https://stdgen.github.io

Let G be a Lie group equipped with commuting left and right linear representations on a vector space V, and let GJoin V be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group GJoin V from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their mathfrak g- and V-components, where mathfrak g is the Lie algebra of G. We also derive the reduced multisymplectic form formula, as well as the Noether and energy-momentum balance laws. Our reduction results are compared with those for principal bundles with standard semidirect-product structure groups, and the role of the centered variables is investigated: they retain the original two-sided geometric data and separate the left and right contributions via the heart and diamond operators. The framework is then applied to a two-sided matrix microstructural model and to the second-order jet group on mathbb R^r, G_r^2. For the matrix model, the conjugation representation decomposes Mat(3) into scalar, vector, and symmetric-traceless sectors, clarifying its precise relation to director and alignment-tensor variables without identifying it with standard nematodynamics. For the one-dimensional jet group on a (1+1)-dimensional Minkowski base, we obtain a semilinear symmetric-hyperbolic system and an exact traveling conversion front with explicit energy-momentum exchange between the linear- and quadratic-jet sectors. Its hyperbolic-target interpretation yields a conserved positive energy for the linearized perturbations. A nonlinear numerical experiment illustrates the reduced evolution and reconstruction of the group-valued field, with consistency assessed through mesh refinement and an independent wave-map computation.

The analysis of non-stationary time-series data requires insight into its local and global patterns with physical interpretability. However, traditional smoothing algorithms, such as B-splines, Savitzky-Golay filtering, and Empirical Mode Decomposition (EMD), lack the ability to perform parametric optimization with guaranteed continuity. In this paper, we propose Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions. By using Levenberg-Marquardt optimization to achieve up to C^1 continuous fitting, FCD transforms raw time-series data into M modes that capture different temporal patterns from short-term to long-term trends. Applications of FCD include physics, medicine, financial analysis, and machine learning, where it is commonly used for the analysis of signal temporal patterns, optimized parameters, derivatives, and integrals of decomposition. Furthermore, FCD can be applied for physical analysis and feature extraction with an average SRMSE of 0.735 per segment and a speed of 0.47s on full decomposition of 1,000 points. Finally, we demonstrate that a Convolutional Neural Network (CNN) enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.

Complex tables with multi-level headers, merged cells and heterogeneous layouts pose persistent challenges for LLMs in both understanding and reasoning. Existing approaches typically rely on table linearization or normalized grid modeling. However, these representations struggle to explicitly capture hierarchical structures and cross-dimensional dependencies, which can lead to misalignment between structural semantics and textual representations for non-standard tables. To address this issue, we propose an Orthogonal Hierarchical Decomposition (OHD) framework that constructs structure-preserving input representations of complex tables for LLMs. OHD introduces an Orthogonal Tree Induction (OTI) method based on spatial--semantic co-constraints, which decomposes irregular tables into a column tree and a row tree to capture vertical and horizontal hierarchical dependencies, respectively. Building on this representation, we design a dual-pathway association protocol to symmetrically reconstruct semantic lineage of each cell, and incorporate an LLM as a semantic arbitrator to align multi-level semantic information. We evaluate OHD framework on two complex table question answering benchmarks, AITQA and HiTab. Experimental results show that OHD consistently outperforms existing representation paradigms across multiple evaluation metrics.

Enabling generative models to decompose visual concepts from a single image is a complex and challenging problem. In this paper, we study a new and challenging task, customized concept decomposition, wherein the objective is to leverage diffusion models to decompose a single image and generate visual concepts from various perspectives. To address this challenge, we propose a two-stage framework, CusConcept (short for Customized Visual Concept Decomposition), to extract customized visual concept embedding vectors that can be embedded into prompts for text-to-image generation. In the first stage, CusConcept employs a vocabulary-guided concept decomposition mechanism to build vocabularies along human-specified conceptual axes. The decomposed concepts are obtained by retrieving corresponding vocabularies and learning anchor weights. In the second stage, joint concept refinement is performed to enhance the fidelity and quality of generated images. We further curate an evaluation benchmark for assessing the performance of the open-world concept decomposition task. Our approach can effectively generate high-quality images of the decomposed concepts and produce related lexical predictions as secondary outcomes. Extensive qualitative and quantitative experiments demonstrate the effectiveness of CusConcept.

Large deep learning models have achieved remarkable success but are resource-intensive, posing challenges such as memory usage. We introduce CURing, a novel model compression method based on CUR matrix decomposition, which approximates weight matrices as the product of selected columns (C) and rows (R), and a small linking matrix (U). We apply this decomposition to weights chosen based on the combined influence of their magnitudes and activations. By identifying and retaining informative rows and columns, CURing significantly reduces model size with minimal performance loss. For example, it reduces Llama3.1-8B's parameters to 7.32B (-9%) in just 129 seconds, over 20 times faster than prior compression methods.

The tensor rank of a bilinear map is the least number of multiplications any bilinear algorithm needs to compute it; for the multiplication of an algebra it measures how cheaply the algebra can be multiplied at all. For the even-dimensional real normed division algebras it is 3 for the complex numbers and 8 for the quaternions, both classical, while for the octonions O only a range was known: at least 15 (Fiduccia and Zalcstein, 1977) and at most 30 (Cariow and Cariowa). We prove $18 le R_{R}(T_{O}) le 25. The lower bound peels the eight slices of T_{O} down to two and bounds the rank of the surviving pencil through the octonion norm. Nothing in it is special to dimension 8: the same steps give R_{R}(T_A) \ge 5{2}n - 2 for every real normed division algebra A of even dimension n, sharp for C and H and the best bound we know for O. The upper bound is a separate construction, an explicit rank-25 decomposition certified by a Krawczyk argument, in exact rational arithmetic, to sit within 10^{-6} of an exact one. The same two arguments pin down the rank of a smaller three-slice quaternion tensor τ, giving R_{R}(τ) = 7$. The Lean 4 kernel checks the lower bounds and the Krawczyk existence principle; the accompanying scripts check the certificate's finitely many exact-rational inequalities.

This work presents Controllable Layer Decomposition (CLD), a method for achieving fine-grained and controllable multi-layer separation of raster images. In practical workflows, designers typically generate and edit each RGBA layer independently before compositing them into a final raster image. However, this process is irreversible: once composited, layer-level editing is no longer possible. Existing methods commonly rely on image matting and inpainting, but remain limited in controllability and segmentation precision. To address these challenges, we propose two key modules: LayerDecompose-DiT (LD-DiT), which decouples image elements into distinct layers and enables fine-grained control; and Multi-Layer Conditional Adapter (MLCA), which injects target image information into multi-layer tokens to achieve precise conditional generation. To enable a comprehensive evaluation, we build a new benchmark and introduce tailored evaluation metrics. Experimental results show that CLD consistently outperforms existing methods in both decomposition quality and controllability. Furthermore, the separated layers produced by CLD can be directly manipulated in commonly used design tools such as PowerPoint, highlighting its practical value and applicability in real-world creative workflows.

While generative models have excelled at creating static 3D content, the pursuit of systems that understand how objects move and respond to interactions remains a fundamental challenge. Current methods for articulated motion lie at a crossroads: they are either physically consistent but too slow for real-time use, or generative but violate basic kinematic constraints. We present DragMesh, a robust framework for real-time interactive 3D articulation built around a lightweight motion generation core. Our core contribution is a novel decoupled kinematic reasoning and motion generation framework. First, we infer the latent joint parameters by decoupling semantic intent reasoning (which determines the joint type) from geometric regression (which determines the axis and origin using our Kinematics Prediction Network (KPP-Net)). Second, to leverage the compact, continuous, and singularity-free properties of dual quaternions for representing rigid body motion, we develop a novel Dual Quaternion VAE (DQ-VAE). This DQ-VAE receives these predicted priors, along with the original user drag, to generate a complete, plausible motion trajectory. To ensure strict adherence to kinematics, we inject the joint priors at every layer of the DQ-VAE's non-autoregressive Transformer decoder using FiLM (Feature-wise Linear Modulation) conditioning. This persistent, multi-scale guidance is complemented by a numerically-stable cross-product loss to guarantee axis alignment. This decoupled design allows DragMesh to achieve real-time performance and enables plausible, generative articulation on novel objects without retraining, offering a practical step toward generative 3D intelligence. Code: https://github.com/AIGeeksGroup/DragMesh. Website: https://aigeeksgroup.github.io/DragMesh.

Scientific archives now contain hundreds of petabytes of data across genomics, ecology, climate, and molecular biology that could reveal undiscovered patterns if systematically analyzed at scale. Large-scale, weakly-supervised datasets in language and vision have driven the development of foundation models whose internal representations encode structure (patterns, co-occurrences and statistical regularities) beyond their training objectives. Most existing methods extract structure only for pre-specified targets; they excel at confirmation but do not support open-ended discovery of unknown patterns. We ask whether sparse autoencoders (SAEs) can enable open-ended feature discovery from foundation model representations. We evaluate this question in controlled rediscovery studies, where the learned SAE features are tested for alignment with semantic concepts on a standard segmentation benchmark and compared against strong label-free alternatives on concept-alignment metrics. Applied to ecological imagery, the same procedure surfaces fine-grained anatomical structure without access to segmentation or part labels, providing a scientific case study with ground-truth validation. While our experiments focus on vision with an ecology case study, the method is domain-agnostic and applicable to models in other sciences (e.g., proteins, genomics, weather). Our results indicate that sparse decomposition provides a practical instrument for exploring what scientific foundation models have learned, an important prerequisite for moving from confirmation to genuine discovery.

Dynamic Mode Decomposition (DMD) is an equation-free method that aims at reconstructing the best linear fit from temporal datasets. In this paper, we show that DMD does not provide accurate approximation for datasets describing oscillatory dynamics, like spiral waves and relaxation oscillations, or spatio-temporal Turing instability. Inspired from the classical "divide and conquer" approach, we propose a piecewise version of DMD (pDMD) to overcome this problem. The main idea is to split the original dataset in N submatrices and then apply the exact (randomized) DMD method in each subset of the obtained partition. We describe the pDMD algorithm in detail and we introduce some error indicators to evaluate its performance when N is increased. Numerical experiments show that very accurate reconstructions are obtained by pDMD for datasets arising from time snapshots of some reaction-diffusion PDE systems, like the FitzHugh-Nagumo model, the lambda-omega system and the DIB morpho-chemical system for battery modeling.

UV unwrapping flattens 3D surfaces to 2D with minimal distortion, often requiring the complex surface to be decomposed into multiple charts. Although extensively studied, existing UV unwrapping methods frequently struggle with AI-generated meshes, which are typically noisy, bumpy, and poorly conditioned. These methods often produce highly fragmented charts and suboptimal boundaries, introducing artifacts and hindering downstream tasks. We introduce PartUV, a part-based UV unwrapping pipeline that generates significantly fewer, part-aligned charts while maintaining low distortion. Built on top of a recent learning-based part decomposition method PartField, PartUV combines high-level semantic part decomposition with novel geometric heuristics in a top-down recursive framework. It ensures each chart's distortion remains below a user-specified threshold while minimizing the total number of charts. The pipeline integrates and extends parameterization and packing algorithms, incorporates dedicated handling of non-manifold and degenerate meshes, and is extensively parallelized for efficiency. Evaluated across four diverse datasets, including man-made, CAD, AI-generated, and Common Shapes, PartUV outperforms existing tools and recent neural methods in chart count and seam length, achieves comparable distortion, exhibits high success rates on challenging meshes, and enables new applications like part-specific multi-tiles packing. Our project page is at https://www.zhaoningwang.com/PartUV.

Simultaneous Localization and Mapping (SLAM) is pivotal in robotics, with photorealistic scene reconstruction emerging as a key challenge. To address this, we introduce Computational Alignment for Real-Time Gaussian Splatting SLAM (CaRtGS), a novel method enhancing the efficiency and quality of photorealistic scene reconstruction in real-time environments. Leveraging 3D Gaussian Splatting (3DGS), CaRtGS achieves superior rendering quality and processing speed, which is crucial for scene photorealistic reconstruction. Our approach tackles computational misalignment in Gaussian Splatting SLAM (GS-SLAM) through an adaptive strategy that optimizes training, addresses long-tail optimization, and refines densification. Experiments on Replica and TUM-RGBD datasets demonstrate CaRtGS's effectiveness in achieving high-fidelity rendering with fewer Gaussian primitives. This work propels SLAM towards real-time, photorealistic dense rendering, significantly advancing photorealistic scene representation. For the benefit of the research community, we release the code on our project website: https://dapengfeng.github.io/cartgs.

The ultimate goal of video generation is to satisfy a fundamental trilemma: achieving high visual quality, maintaining rigorous physical consistency, and enabling precise controllability. While recent models can maintain this balance in simple, isolated scenarios, we observe that this equilibrium is fragile and often breaks down as scene complexity increases (e.g., involving collisions or dense traffic). To address this, we introduce Motion Forcing, a framework designed to stabilize this trilemma even in complex generative tasks. Our key insight is to explicitly decouple physical reasoning from visual synthesis via a hierarchical ``Point-Shape-Appearance'' paradigm. This approach decomposes generation into verifiable stages: modeling complex dynamics as sparse geometric anchors (Point), expanding them into dynamic depth maps that explicitly resolve 3D geometry (Shape), and finally rendering high-fidelity textures (Appearance). Furthermore, to foster robust physical understanding, we employ a Masked Point Recovery strategy. By randomly masking input anchors during training and enforcing the reconstruction of complete dynamic depth, the model is compelled to move beyond passive pattern matching and learn latent physical laws (e.g., inertia) to infer missing trajectories. Extensive experiments on autonomous driving benchmarks show that Motion Forcing significantly outperforms state-of-the-art baselines, maintaining trilemma stability across complex scenes. Evaluations on physics and robotics further confirm our framework's generality.

Decomposing model activations into interpretable components is a key open problem in mechanistic interpretability. Sparse autoencoders (SAEs) are a popular method for decomposing the internal activations of trained transformers into sparse, interpretable features, and have been applied to MLP layers and the residual stream. In this work we train SAEs on attention layer outputs and show that also here SAEs find a sparse, interpretable decomposition. We demonstrate this on transformers from several model families and up to 2B parameters. We perform a qualitative study of the features computed by attention layers, and find multiple families: long-range context, short-range context and induction features. We qualitatively study the role of every head in GPT-2 Small, and estimate that at least 90% of the heads are polysemantic, i.e. have multiple unrelated roles. Further, we show that Sparse Autoencoders are a useful tool that enable researchers to explain model behavior in greater detail than prior work. For example, we explore the mystery of why models have so many seemingly redundant induction heads, use SAEs to motivate the hypothesis that some are long-prefix whereas others are short-prefix, and confirm this with more rigorous analysis. We use our SAEs to analyze the computation performed by the Indirect Object Identification circuit (Wang et al.), validating that the SAEs find causally meaningful intermediate variables, and deepening our understanding of the semantics of the circuit. We open-source the trained SAEs and a tool for exploring arbitrary prompts through the lens of Attention Output SAEs.

In this study, we examine the representation learning abilities of Denoising Diffusion Models (DDM) that were originally purposed for image generation. Our philosophy is to deconstruct a DDM, gradually transforming it into a classical Denoising Autoencoder (DAE). This deconstructive procedure allows us to explore how various components of modern DDMs influence self-supervised representation learning. We observe that only a very few modern components are critical for learning good representations, while many others are nonessential. Our study ultimately arrives at an approach that is highly simplified and to a large extent resembles a classical DAE. We hope our study will rekindle interest in a family of classical methods within the realm of modern self-supervised learning.

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

We present an algorithmic framework for incremental maintenance of first sheaf cohomology H^1(X; F) on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of H^1 via factorization of the coboundary matrix requires O(n^3) time; when the complex evolves with a stream of m edits, full recomputation after each edit costs O(mn^3). Under a bounded local geometry assumption -- bounded cell size v_{max}, bounded stalk dimension d, and bounded nerve degree D -- each edit (vertex insertion, edge insertion, restriction map update) affects only a bounded set of local coboundary blocks. The algorithm therefore processes lazy streaming edits in O(1) time with respect to the total complex size n (with cost polynomial in the local geometry parameters v_{max}, d, and D, which are treated as constants independent of n), deferring local eigensolves and Mayer-Vietoris global assembly to synchronization points (Flush). At synchronization, the maintained state agrees with the corresponding batch assembly of the partitioned sheaf model; we observe zero measured drift in all batch-verified runs (through V = 10^6). We also give an amortized O(|E|) streaming construction for the cellular decomposition and discuss an adversarial algebraic-RAM barrier arguing that unpartitioned non-trivial sheaves (d geq 2, non-identity restriction maps) do not admit the same locality. Experiments on Barabasi-Albert graphs with up to 5 times 10^6 vertices and 1.7 times 10^7 streaming edits show 35 μs median lazy per-edit update latency (excluding flush); query time (global assembly at synchronization) is O(n) per flush in the implemented full-traversal path. Exact synchronization costs are reported separately.

In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this to topological concepts such as homeomorphism and embedding. We then investigate what are continuous and discontinuous representations for 2D, 3D, and n-dimensional rotations. We demonstrate that for 3D rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions. Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. We also present continuous representations for the general case of the n-dimensional rotation group SO(n). While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms. We finally present empirical results, which show that our continuous rotation representations outperform discontinuous ones for several practical problems in graphics and vision, including a simple autoencoder sanity test, a rotation estimator for 3D point clouds, and an inverse kinematics solver for 3D human poses.

We establish explicit means via which natural dilations of completely positive (CP) maps can be constructed à la Kraus's IInd representation theorem. To obtain this, we rely on the Choi-Jamiołkowski correspondence and develop a Cholesky algorithm for bi-partite systems. This enables a canonical construction of adjoint actions which recover the behaviour of the original CP-maps. Our results hold under separability assumptions and the requirement that the maps are completely bounded and preserve the subideal of finite rank operators.

Decomposing a scene into its shape, reflectance, and illumination is a challenging but important problem in computer vision and graphics. This problem is inherently more challenging when the illumination is not a single light source under laboratory conditions but is instead an unconstrained environmental illumination. Though recent work has shown that implicit representations can be used to model the radiance field of an object, most of these techniques only enable view synthesis and not relighting. Additionally, evaluating these radiance fields is resource and time-intensive. We propose a neural reflectance decomposition (NeRD) technique that uses physically-based rendering to decompose the scene into spatially varying BRDF material properties. In contrast to existing techniques, our input images can be captured under different illumination conditions. In addition, we also propose techniques to convert the learned reflectance volume into a relightable textured mesh enabling fast real-time rendering with novel illuminations. We demonstrate the potential of the proposed approach with experiments on both synthetic and real datasets, where we are able to obtain high-quality relightable 3D assets from image collections. The datasets and code is available on the project page: https://markboss.me/publication/2021-nerd/

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

Most scientific materials compress reasoning, presenting conclusions while omitting the derivational chains that justify them. This compression hinders verification by lacking explicit, step-wise justifications and inhibits cross-domain links by collapsing the very pathways that establish the logical and causal connections between concepts. We introduce a scalable framework that decompresses scientific reasoning, constructing a verifiable Long Chain-of-Thought (LCoT) knowledge base and projecting it into an emergent encyclopedia, SciencePedia. Our pipeline operationalizes an endpoint-driven, reductionist strategy: a Socratic agent, guided by a curriculum of around 200 courses, generates approximately 3 million first-principles questions. To ensure high fidelity, multiple independent solver models generate LCoTs, which are then rigorously filtered by prompt sanitization and cross-model answer consensus, retaining only those with verifiable endpoints. This verified corpus powers the Brainstorm Search Engine, which performs inverse knowledge search -- retrieving diverse, first-principles derivations that culminate in a target concept. This engine, in turn, feeds the Plato synthesizer, which narrates these verified chains into coherent articles. The initial SciencePedia comprises approximately 200,000 fine-grained entries spanning mathematics, physics, chemistry, biology, engineering, and computation. In evaluations across six disciplines, Plato-synthesized articles (conditioned on retrieved LCoTs) exhibit substantially higher knowledge-point density and significantly lower factual error rates than an equally-prompted baseline without retrieval (as judged by an external LLM). Built on this verifiable LCoT knowledge base, this reasoning-centric approach enables trustworthy, cross-domain scientific synthesis at scale and establishes the foundation for an ever-expanding encyclopedia.

In this paper, we explore the use of a variational autoencoder (VAE), a deep generative model, to compress and generate images of dark matter density fields from ΛCDM like cosmological simulations. The VAE learns a compact, low-dimensional representation of the large-scale structure, enabling both accurate reconstruction and generation of statistically realistic samples. We evaluated the generated images by comparing their power spectra to those of real simulations and the theoretical ΛCDM prediction, finding strong agreement with the state-of-the-art simulations. In addition, the VAE provides a fast and scalable method for generating synthetic cosmological data, making it a valuable tool for data augmentation. These capabilities can accelerate the development and training of more advanced machine learning models for cosmological analysis, particularly in scenarios where large-scale simulations are computationally expensive. Our results highlight the potential for generative artificial intelligence as a practical bridge between physical modeling and modern deep learning in cosmology.

Using a set of LambdaCDM simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art tools of multiscale topological data analysis (TDA). We follow the development of the cosmic web topology in terms of the evolution of Betti number curves and feature persistence diagrams of the three (topological) classes of structural features: matter concentrations, filaments and tunnels, and voids. The Betti curves specify the prominence of features as a function of density level, and their evolution with cosmic epoch reflects the changing network connections between these structural features. The persistence diagrams quantify the longevity and stability of topological features. In this study we establish, for the first time, the link between persistence diagrams, the features they show, and the gravitationally driven cosmic structure formation process. By following the diagrams' development over cosmic time, the link between the multiscale topology of the cosmic web and the hierarchical buildup of cosmic structure is established. The sharp apexes in the diagrams are intimately related to key transitions in the structure formation process. The apex in the matter concentration diagrams coincides with the density level at which, typically, they detach from the Hubble expansion and begin to collapse. At that level many individual islands merge to form the network of the cosmic web and a large number of filaments and tunnels emerge to establish its connecting bridges. The location trends of the apex possess a self-similar character that can be related to the cosmic web's hierarchical buildup. We find that persistence diagrams provide a significantly higher and more profound level of information on the structure formation process than more global summary statistics like Euler characteristic or Betti numbers.

Cooperative multi-agent reinforcement learning (MARL) is widely used to address large joint observation and action spaces by decomposing a centralized control problem into multiple interacting agents. However, such decomposition often introduces additional challenges, including non-stationarity, unstable training, weak coordination, and limited theoretical guarantees. In this paper, we propose the Consensus Multi-Agent Transformer (CMAT), a centralized framework that bridges cooperative MARL to a hierarchical single-agent reinforcement learning (SARL) formulation. CMAT treats all agents as a unified entity and employs a Transformer encoder to process the large joint observation space. To handle the extensive joint action space, we introduce a hierarchical decision-making mechanism in which a Transformer decoder autoregressively generates a high-level consensus vector, simulating the process by which agents reach agreement on their strategies in latent space. Conditioned on this consensus, all agents generate their actions simultaneously, enabling order-independent joint decision making and avoiding the sensitivity to action-generation order in conventional Multi-Agent Transformers (MAT). This factorization allows the joint policy to be optimized using single-agent PPO while preserving expressive coordination through the latent consensus. To evaluate the proposed method, we conduct experiments on benchmark tasks from StarCraft II, Multi-Agent MuJoCo, and Google Research Football. The results show that CMAT achieves superior performance over recent centralized solutions, sequential MARL methods, and conventional MARL baselines. The code for this paper is available at:https://github.com/RS2002/CMAT .

For finite nilpotent groups J and N, suppose J acts on N via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow p-subgroups of J that mirrors the primary decomposition of H^1(J,N) for abelian N. We then show that if N rtimes J acts on some non-empty set Omega, where the action of N is transitive and for each prime p a Sylow p-subgroup of J fixes an element of Omega, then J fixes an element of Omega.

Reconstructing the early universe from the evolved present-day universe is a challenging and computationally demanding problem in modern astrophysics. We devise a novel generative framework, Cosmo3DFlow, designed to address dimensionality and sparsity, the critical bottlenecks inherent in current state-of-the-art methods for cosmological inference. By integrating 3D Discrete Wavelet Transform (DWT) with flow matching, we effectively represent high-dimensional cosmological structures. The Wavelet Transform addresses the ``void problem'' by translating spatial emptiness into spectral sparsity. It decouples high-frequency details from low-frequency structures, and wavelet-space velocity fields facilitate stable ordinary differential equation (ODE) solvers with large step sizes. Using large-scale cosmological N-body simulations at 128^3 resolution, we achieve up to 46times faster sampling than diffusion models. Our results enable initial conditions to be sampled in seconds, compared to minutes for previous methods.

We present a computational framework that transforms single images into 3D physical objects. The visual geometry of a physical object in an image is determined by three orthogonal attributes: mechanical properties, external forces, and rest-shape geometry. Existing single-view 3D reconstruction methods often overlook this underlying composition, presuming rigidity or neglecting external forces. Consequently, the reconstructed objects fail to withstand real-world physical forces, resulting in instability or undesirable deformation -- diverging from their intended designs as depicted in the image. Our optimization framework addresses this by embedding physical compatibility into the reconstruction process. We explicitly decompose the three physical attributes and link them through static equilibrium, which serves as a hard constraint, ensuring that the optimized physical shapes exhibit desired physical behaviors. Evaluations on a dataset collected from Objaverse demonstrate that our framework consistently enhances the physical realism of 3D models over existing methods. The utility of our framework extends to practical applications in dynamic simulations and 3D printing, where adherence to physical compatibility is paramount.

Modern scene reconstruction methods, such as 3D Gaussian Splatting, deliver photo-realistic novel view synthesis at real-time speeds, yet their adoption in interactive graphics applications has been limited. A major bottleneck is the difficulty of interacting with these representations compared to traditional, human-authored 3D assets. While previous research has attempted to impose semantic decomposition on these models, significant challenges remain regarding segmentation quality and consistency. To address this, we introduce Semantic Foam, extending the recently proposed Radiant Foam representations to semantic decomposition tasks. Our approach integrates the natural spatial volumetric decomposition of Radiant Foam's Voronoi mesh with an explicit semantic feature field parameterized at the cell level. This explicit structure enables direct spatial regularization, which prevents artifacts caused by occlusion or inconsistent supervision across views - common pitfalls for other point-based representations. Experimental results show that our method achieves comparable or superior object-level segmentation performance compared to state-of-the-art methods like Gaussian Grouping and SAGA.

We study the problem of cooperative multi-agent reinforcement learning with a single joint reward signal. This class of learning problems is difficult because of the often large combined action and observation spaces. In the fully centralized and decentralized approaches, we find the problem of spurious rewards and a phenomenon we call the "lazy agent" problem, which arises due to partial observability. We address these problems by training individual agents with a novel value decomposition network architecture, which learns to decompose the team value function into agent-wise value functions. We perform an experimental evaluation across a range of partially-observable multi-agent domains and show that learning such value-decompositions leads to superior results, in particular when combined with weight sharing, role information and information channels.

This paper presents algorithms for hierarchical, agglomerative clustering which perform most efficiently in the general-purpose setup that is given in modern standard software. Requirements are: (1) the input data is given by pairwise dissimilarities between data points, but extensions to vector data are also discussed (2) the output is a "stepwise dendrogram", a data structure which is shared by all implementations in current standard software. We present algorithms (old and new) which perform clustering in this setting efficiently, both in an asymptotic worst-case analysis and from a practical point of view. The main contributions of this paper are: (1) We present a new algorithm which is suitable for any distance update scheme and performs significantly better than the existing algorithms. (2) We prove the correctness of two algorithms by Rohlf and Murtagh, which is necessary in each case for different reasons. (3) We give well-founded recommendations for the best current algorithms for the various agglomerative clustering schemes.

Given a factorization of an image into a sum of linear components, we present a zero-shot method to control each individual component through diffusion model sampling. For example, we can decompose an image into low and high spatial frequencies and condition these components on different text prompts. This produces hybrid images, which change appearance depending on viewing distance. By decomposing an image into three frequency subbands, we can generate hybrid images with three prompts. We also use a decomposition into grayscale and color components to produce images whose appearance changes when they are viewed in grayscale, a phenomena that naturally occurs under dim lighting. And we explore a decomposition by a motion blur kernel, which produces images that change appearance under motion blurring. Our method works by denoising with a composite noise estimate, built from the components of noise estimates conditioned on different prompts. We also show that for certain decompositions, our method recovers prior approaches to compositional generation and spatial control. Finally, we show that we can extend our approach to generate hybrid images from real images. We do this by holding one component fixed and generating the remaining components, effectively solving an inverse problem.

Large Reasoning Models (LRMs) are criticized for the excessively lengthy Chain-of-Thought (CoT) to derive the final answer, suffering from high first-token and overall latency. Typically, the CoT of LRMs mixes multiple thinking units; each unit attempts to produce a candidate answer to the original query. Hence, a natural idea to improve efficiency is to reduce the unit number. Yet, the fact that the thinking units in vanilla CoT cannot be explicitly managed renders doing so challenging. This paper introduces Multi-Turn Decomposition (MinD) to decode conventional CoT into a sequence of explicit, structured, and turn-wise interactions to bridge the gap. In MinD, the model provides a multi-turn response to the query, where each turn embraces a thinking unit and yields a corresponding answer. The subsequent turns can reflect, verify, revise, or explore alternative approaches to both the thinking and answer parts of earlier ones. This not only makes the answer delivered more swiftly, but also enables explicit controls over the iterative reasoning process (i.e., users may halt or continue at any turn). We follow a supervised fine-tuning (SFT) then reinforcement learning (RL) paradigm to realize MinD. We first rephrase the outputs of an LRM into multi-turn formats by prompting another LLM, and then tune the LRM with such data. Observing that the tuned model tends to consume even more tokens than the original one (probably due to that the multi-turn formats introduce additional answer tokens), we advocate leveraging RL algorithms like GRPO to prioritize correct outputs with fewer turns. Trained on the MATH dataset using R1-Distill models, MinD can achieve up to ~70% reduction in both output token usage and time to first token (TTFT), while maintaining competitive performance on reasoning benchmarks such as MATH-500, AIME24, AMC23, and GPQA-Diamond.

LLMs have achieved remarkable breakthroughs in reasoning, insights, and tool use, but chaining these abilities into extended processes at the scale of those routinely executed by humans, organizations, and societies has remained out of reach. The models have a persistent error rate that prevents scale-up: for instance, recent experiments in the Towers of Hanoi benchmark domain showed that the process inevitably becomes derailed after at most a few hundred steps. Thus, although LLM research is often still benchmarked on tasks with relatively few dependent logical steps, there is increasing attention on the ability (or inability) of LLMs to perform long range tasks. This paper describes MAKER, the first system that successfully solves a task with over one million LLM steps with zero errors, and, in principle, scales far beyond this level. The approach relies on an extreme decomposition of a task into subtasks, each of which can be tackled by focused microagents. The high level of modularity resulting from the decomposition allows error correction to be applied at each step through an efficient multi-agent voting scheme. This combination of extreme decomposition and error correction makes scaling possible. Thus, the results suggest that instead of relying on continual improvement of current LLMs, massively decomposed agentic processes (MDAPs) may provide a way to efficiently solve problems at the level of organizations and societies.

I propose a Hamiltonian framework for chiral gauge theories (CGT) based on a Euclidean formulation which uses 2n dimensional chiral fermions on the boundary of a 2n+1 dimensional disk. In the original Euclidean formulation, boundary gauge fields were extended into the bulk using 2n + 1 dimensional gauge field equations of motion (EOM) which creates a bottleneck for constructing a Hamiltonian. I present an alternate proposal for extending the gauge fields into the bulk that is compatible with both a Hamiltonian framework and a Euclidean path integral. Applying this to Abelian gauge fields produces exact expressions of interior fields as a functional of the boundary fields, which can be directly used in a Hamiltonian formulation. Euclidean analysis of the new gauge field extension shows that it can preserve the non-perturbative content of the original construction, including the behavior of topological charge, associated chiral fermion zero modes and an absence of the strong CP problem when applied to the Standard Model. This construction opens up a route to a Hamiltonian treatment and future quantum simulation of CGTs on a disk boundary.

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group G acts transitively on a space M, any G-invariant function on a product space X times M can be reduced to an invariant of the isotropy subgroup H of M acting on X alone. Our approach establishes an explicit orbit equivalence (X times M)/G cong X/H, yielding a principled reduction that preserves expressivity. We apply this characterization to Equivariant Neural Fields, extending them to arbitrary group actions and homogeneous conditioning spaces, and thereby removing the major structural constraints imposed by existing methods.

Intrinsic decomposition is a fundamental mid-level vision problem that plays a crucial role in various inverse rendering and computational photography pipelines. Generating highly accurate intrinsic decompositions is an inherently under-constrained task that requires precisely estimating continuous-valued shading and albedo. In this work, we achieve high-resolution intrinsic decomposition by breaking the problem into two parts. First, we present a dense ordinal shading formulation using a shift- and scale-invariant loss in order to estimate ordinal shading cues without restricting the predictions to obey the intrinsic model. We then combine low- and high-resolution ordinal estimations using a second network to generate a shading estimate with both global coherency and local details. We encourage the model to learn an accurate decomposition by computing losses on the estimated shading as well as the albedo implied by the intrinsic model. We develop a straightforward method for generating dense pseudo ground truth using our model's predictions and multi-illumination data, enabling generalization to in-the-wild imagery. We present an exhaustive qualitative and quantitative analysis of our predicted intrinsic components against state-of-the-art methods. Finally, we demonstrate the real-world applicability of our estimations by performing otherwise difficult editing tasks such as recoloring and relighting.

The decompose-then-verify strategy for verification of Large Language Model (LLM) generations decomposes claims that are then independently verified. Decontextualization augments text (claims) to ensure it can be verified outside of the original context, enabling reliable verification. While decomposition and decontextualization have been explored independently, their interactions in a complete system have not been investigated. Their conflicting purposes can create tensions: decomposition isolates atomic facts while decontextualization inserts relevant information. Furthermore, a decontextualized subclaim presents a challenge to the verification step: what part of the augmented text should be verified as it now contains multiple atomic facts? We conduct an evaluation of different decomposition, decontextualization, and verification strategies and find that the choice of strategy matters in the resulting factuality scores. Additionally, we introduce DnDScore, a decontextualization aware verification method which validates subclaims in the context of contextual information.

The popularity of large-scale pre-training has promoted the development of medical foundation models. However, some studies have shown that although foundation models exhibit strong general feature extraction capabilities, their performance on specific tasks is still inferior to task-specific methods. In this paper, we explore a new perspective called ``Knowledge Decomposition'' to improve the performance on specific medical tasks, which deconstruct the foundation model into multiple lightweight expert models, each dedicated to a particular task, with the goal of improving specialization while concurrently mitigating resource expenditure. To accomplish the above objective, we design a novel framework named Low-Rank Knowledge Decomposition (LoRKD), which explicitly separates graidents by incorporating low-rank expert modules and the efficient knowledge separation convolution. Extensive experimental results demonstrate that the decomposed models perform well in terms of performance and transferability, even surpassing the original foundation models.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

Structural fidelity is essential to scientific methodology diagrams. To communicate research logic, these diagrams must faithfully render components, directional relations, and textual annotations. Since a single error, such as a reversed arrow or an unreadable equation, can invalidate the entire figure, structural fidelity is inherently conjunctive: correctness on one axis cannot compensate for failure on another. Current open-source models fail to satisfy this criterion. Supervised fine-tuning (SFT) learns plausible layouts but cannot reliably ensure structural correctness, while scalar reward-based post-training obscures which structural dimension has failed. To address this, we introduce SciForma, a framework for the structure faithful generation of scientific methodology diagrams. Specifically, SciForma decomposes diagram quality into three structural axes: Component, Arrow, and Text, guided by a structural inventory. Built on this foundation, we curate SciFormaData-700K for structured training and SciFormaBench-2K for logic-verified evaluation. To close the gap left by SFT, we develop Multi-Dimensional Conjunctive Preference Optimization (M-DPO), which enforces simultaneous correctness across all axes and adaptively routes gradients to the most deficient dimension in post-training. The same structural inventory also enables iterative editing at inference time to correct residual errors. This combination allows SciForma-9B to exceed all open-source baselines and GPT-Image-1.5 on both SciFormaBench-2K and AIBench, bringing open scientific diagram generation close to proprietary-level structural fidelity. Our code and data will be available at: https://github.com/microsoft/SciForma.