The paper analyzes how grid cells perform path integration and learns hexagon grid patterns.
problem Understanding how grid cells perform path integration calculations.
method Theoretical analysis of a general representation model of path integration by grid cells, identifying group representation and isotropic scaling conditions.
result The learned model of hexagon grid patterns is capable of accurate long distance path integration.
The paper studies how grid cell patterns emerge in neural networks.
problem Understanding how grid cells in the brain form hexagonal firing patterns.
method Training recurrent neural networks with conformal normalization of velocity inputs.
result Conformal normalization is crucial for the emergence of hexagonal grid patterns in neural networks.
This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
Existing techniques to compress point cloud attributes leverage either geometric or video-based compression tools. We explore a radically different approach inspired by recent advances in point cloud representation learning. Point clouds can be interpreted as 2D manifolds in 3D space. Specifically, we fold a 2D grid on…
Study freezing sets for digital images in a 2D grid.
problem Determine minimal freezing sets for digital images.
method Prove methods to obtain freezing sets for digital images (X, c_i) where X is a subset of Z^2.
result Examples show how methods can lead to the determination of minimal freezing sets.
Computes elastic grids that approximate 3D surfaces without physical simulations.
problem Creating planar grids that fit complex 3D surfaces efficiently.
method Uses differential geometry to minimize bending energy and nestle to the surface.
result Elastic grids can approximate 3D surfaces without physical simulations.
2D path planning in static environment is a well-known problem and one of the common ways to solve it is to 1) represent the environment as a grid and 2) perform a heuristic search for a path on it. At the same time 2D grid resembles much a digital image, thus an appealing idea comes to being -- to treat the problem as…
Deep learning speeds spectral density estimation for large 2D/3D grids.
problem Computational challenges in estimating spectral densities for large grids.
method Deep learning neural network for spectral density estimation.
result Deep learning estimator is a universal approximator and faster than existing methods.
Designs a Cellular Automata rule for forming touching loop patterns.
problem Forming stable touching loop patterns in a 2D grid.
method Developed a Cellular Automata rule that uses templates to cover the space and match patterns.
result The rule successfully evolves stable touching loop patterns in a 2D grid.
This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The mode…
Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.
problem Understanding the algebraic, geometric, and topological properties of grid cells.
method Investigating recurrent neural network models of grid cells, focusing on Lie group and Lie algebra representations, conformal isometry, and hexagon periodic patterns.
result Conformal isometry leads to hexagon periodic patterns in grid cell responses and accurate path integration.
Logistic Gaussian process (LGP) priors provide a flexible alternative for modelling unknown densities. The smoothness properties of the density estimates can be controlled through the prior covariance structure of the LGP, but the challenge is the analytically intractable inference. In this paper, we present approximat…
Decades of research on the neural code underlying spatial navigation have revealed a diverse set of neural response properties. The Entorhinal Cortex (EC) of the mammalian brain contains a rich set of spatial correlates, including grid cells which encode space using tessellating patterns. However, the mechanisms and fu…
3D RadViz improves 3D data visualization of multidimensional datasets.
problem Tackles the challenge of visualizing multidimensional datasets in 3D.
method Develops RadViz3D, a 3D radial visualization tool with uniform anchor points.
result Improves the display of multidimensional datasets, especially for uncorrelated variables.
The uninformative ordering of artificial neurons in Deep Neural Networks complicates visualizing activations in deeper layers. This is one reason why the internal structure of such models is very unintuitive. In neuroscience, activity of real brains can be visualized by highlighting active regions. Inspired by those te…
pLSTM tackles long-range language modeling and computer vision tasks with parallelizable linear source transition mark networks.
problem Challenges of existing recurrent architectures in handling sequences and multi-dimensional data.
method Introduces pLSTM, a parallelizable linear source transition mark network for linear graphs and DAGs, addressing vanishing/exploding activation/gradient issues.
result pLSTM outperforms Transformers in long-range tasks like arrow-pointing extrapolation and image size extrapolation.
In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher acc…
The Poisson equation is commonly encountered in engineering, for instance in computational fluid dynamics (CFD) where it is needed to compute corrections to the pressure field to ensure the incompressibility of the velocity field. In the present work, we propose a novel fully convolutional neural network (CNN) architec…
Assisted by neural networks, reinforcement learning agents have been able to solve increasingly complex tasks over the last years. The simulation environment in which the agents interact is an essential component in any reinforcement learning problem. The environment simulates the dynamics of the agents' world and henc…
The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If G is an infinite graph with cofinite free Zd-symmetry, then the logarithmic Mahler measure m(Δ) of its Laplacian polynomial Δ is the …
Deep learning upscales geologic models efficiently.
problem Upscaling large-scale geologic models for efficient simulation.
method Theory-guided convolutional neural network (TgCNN) trained to approximate hydraulic conductivity relationships.
result Deep learning method achieves equivalent upscaling accuracy to numerical methods but with significantly improved efficiency.
The paper develops estimators for variance in graph structures using fused lasso.
problem Variance estimation in graph-structured problems.
method Developed linear time estimator for homoscedastic case and total variation regularization estimator for heteroscedastic case.
result Minimax rates and consistency for variance estimation in various graph structures.
We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…
In compressed sensing MRI (CS-MRI), k-space measurements are under-sampled to achieve accelerated scan times. CS-MRI presents two fundamental problems: (1) where to sample and (2) how to reconstruct an under-sampled scan. In this paper, we tackle both problems simultaneously for the specific case of 2D Cartesian sampli…
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in CP2 under certain conditions. Existing attention mechanisms are trained to attend to individual items in a collection (the memory) with a predefined, fixed granularity, e.g., a word token or an image grid. We propose area attention: a way to attend to areas in the memory, where each area contains a group of items that are structurally adjacent, e.g…
We propose a targeted communication architecture for multi-agent reinforcement learning, where agents learn both what messages to send and whom to address them to while performing cooperative tasks in partially-observable environments. This targeting behavior is learnt solely from downstream task-specific reward withou…
2D CNNs approximate Korobov functions with near-optimal rates.
problem Approximating Korobov functions using 2D CNNs.
method Constructive approach for 2D CNNs with ReLU activations and fully connected layers.
result 2D CNNs achieve near-optimal approximation rates for Korobov functions.
Half grid diagrams prove every link can be represented by a special type of grid diagram.
problem Representing links using grid diagrams and related invariants.
method Defining half grid diagrams and constructing canonical pairs, proving equivalence to Jones' construction, relating to classical link invariants.
result Established a new method to relate the oriented Thompson index to classical link invariants and provided bounds for knot invariants.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
GridPyM handles grid diagrams for knot theory.
problem Handling grid diagrams for knot theory.
method Generates and simplifies grids, models local transformations.
result Models local transformations between grid diagrams.
Grid homology theory for spatial graphs extends skein sequence.
problem No specific problem stated; focuses on extending a sequence.
method Defined grid homology theory for spatial graphs and extended skein sequence.
result Skein exact sequence extended to grid homology for spatial graphs.
Extends knot invariant to filtered grid complexes.
problem Knot invariants and grid complexes.
method Combining Ozsváth-Szabó-Stipsicz crossing-change maps with Alishahi-Eftekhary l(K) invariant.
result Combinatorial formulation of knot invariant.
New method finds grid diagrams for many fibered knots.
problem Detecting fibered knots using grid diagrams.
method Developed an efficient method to identify grid diagrams with unique maximal Alexander grading states.
result Found suitable grid diagrams for 5385 of 5397 fibered prime knots with crossing number ≤ 13.
Grid homology properties for MOY graphs studied.
problem Defining and studying properties of grid homology for MOY graphs.
method Defined grid homology from Harvey and O'Donnol's work. Studied properties using oriented skein relation, edge contraction, and parallel edge unification.
result Properties of grid homology for MOY graphs were studied and defined.
Grid homology invariant proved for lens space links.
problem Proving combinatorial invariance of grid homology for lens space links.
method Combining combinatorial methods with sign assignments to prove invariance.
result Grid homology is a link invariant for lens space links.
New trading strategy beats traditional grid in crypto markets.
problem Low expected return of traditional grid trading strategy.
method Dynamic Grid Trading (DGT) strategy that adapts to market conditions.
result DGT strategy outperforms traditional grid and buy-and-hold strategies.
New method constructs moduli spaces of Lagrangian surfaces in CP^2 from grid diagrams.
problem Constructing explicit examples of triple grid diagrams for Lagrangian surfaces in CP^2.
method Elegant geometric construction reducing to linear algebra.
result Explicit construction of moduli space of triple grid diagrams.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
Grid homology shows knot unknotting lower bound.
problem Knot unknotting number determination
method Grid homology analysis
result Torsion homology classes order bounds unknotting number
Computes homology of an obstruction chain complex in grid homology.
problem Computing the homology of an obstruction chain complex in grid homology.
method Defined and computed the homology of the obstruction chain complex of the full grid.
result Results about the existence of sign assignments in grid homology.
SKI accelerates GP inference with sparse grids to handle higher dimensions.
problem SKI scales poorly in high dimensions due to dense grid size.
method Sparse grids within SKI framework, novel matrix-vector multiplication algorithm.
result SKI can be scaled to higher dimensions while maintaining accuracy.
Proposes a model to generate 3D-aware images from 2D images.
problem Generating 3D-aware images from 2D images.
method Likelihood-based top-down model using Neural Radiance Fields and energy-based latent variables.
result Model can infer 3D object structures from 2D images and generate novel views.
In this paper we propose a new method to predict the final destination of vehicle trips based on their initial partial trajectories. We first review how we obtained clustering of trajectories that describes user behaviour. Then, we explain how we model main traffic flow patterns by a mixture of 2d Gaussian distribution…
Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
problem Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
method Direct computations
result Derive local bihamiltonian structure
Enhances 2D face recognition with 3D features using active illumination.
problem Improving robustness of 2D face recognition to spoofing attacks and low-light conditions.
method Projecting a high spatial frequency pattern onto the face to recover 3D information and a 2D image simultaneously.
result Significantly boosts face recognition performance and dramatically improves robustness to spoofing attacks.
The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.
problem Understanding grid homology for spatial graphs with various types of edges.
method Developed grid homology for spatial graphs with cut edges and applied it to prove a Künneth formula for connected sums.
result A Künneth formula for knot Floer homology of connected sums is proven using grid homology.
Minimal grid diagrams for 12-crossing prime knots identified.
problem Identifying minimal grid diagrams for prime knots.
method Listed minimal grid diagrams for 12-crossing prime knots.
result Provided a list of minimal grid diagrams for 12-crossing prime knots.