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.
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48 results for “mapping grid”
A novel method compresses point cloud attributes by folding them onto a 2D grid.
problem Efficiently compressing point cloud attributes for storage and transmission.
method Interpreting point clouds as 2D manifolds, folding onto a grid, and mapping attributes to the grid using optimized methods.
result The proposed folding-based approach achieves performance comparable to state-of-the-art codecs.
We give some general criteria of being a homeomorphism for continuous mappings of topological manifolds, as well as criteria of being a diffeomorphism for smooth mappings of smooth manifolds. As an illustration, we apply these criteria to the problems arising in two- and three-dimensional grid generation.
A Note on Topology Preservation in Classification, and the Construction of a Universal Neuron Gridcs.NE
It will be shown that according to theorems of K. Menger, every neuron grid if identified with a curve is able to preserve the adopted qualitative structure of a data space. Furthermore, if this identification is made, the neuron grid structure can always be mapped to a subset of a universal neuron grid which is constr…
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.
Insights in power grid pixel maps (PGPMs) refer to important facility operating states and unexpected changes in the power grid. Identifying insights helps analysts understand the collaboration of various parts of the grid so that preventive and correct operations can be taken to avoid potential accidents. Existing sol…
Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalizationq-bio.NC
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.
We survey agglomerative hierarchical clustering algorithms and discuss efficient implementations that are available in R and other software environments. We look at hierarchical self-organizing maps, and mixture models. We review grid-based clustering, focusing on hierarchical density-based approaches. Finally we descr…
WindDragon forecasts wind power with deep learning.
problem Accurate short-term wind power forecasting is crucial for grid operation.
method Automated Deep Learning combined with Numerical Weather Predictions.
result Automated Deep Learning improves wind power forecasting accuracy.
Multi-Stage Fault Warning for Large Electric Grids Using Anomaly Detection and Machine Learningcs.LG
In the monitoring of a complex electric grid, it is of paramount importance to provide operators with early warnings of anomalies detected on the network, along with a precise classification and diagnosis of the specific fault type. In this paper, we propose a novel multi-stage early warning system prototype for electr…
Node2Grids uncouples GCN training for large graphs, saving memory and computation.
problem GCNs' coupled training framework limits flexibility and scalability for large-scale graphs.
method Node2Grids maps coupled graph data into independent grid-like data for efficient processing.
result Node2Grids achieves comparable results to GCNs while saving memory and computation.
Efficiently prices American options with multiple assets using sparse grids.
problem Pricing American options with multiple underlying assets efficiently.
method Dynamic programming formulation followed by sparse grid interpolation.
result Sparse grids reduce the number of interpolation points and maintain function smoothness.
Bayesian deep learning improves geostatistical mapping with auxiliary data.
problem Traditional geostatistical methods are limited in feature learning and uncertainty estimation.
method Deep neural networks learn complex relationships from auxiliary data for probabilistic mapping.
result Deep learning produces detailed, probabilistic maps with uncertainty estimates.
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.
New method characterizes surface quadrilateral layouts as special immersions.
problem Characterize surface quadrilateral layouts mathematically.
method Characterizes quadrilateral layouts as special immersions of a cut representation of the surface into the Euclidean plane.
result Mathematically describes and generalizes integer grid maps.
We present a new method of learning a continuous occupancy field for use in robot navigation. Occupancy grid maps, or variants of, are possibly the most widely used and accepted method of building a map of a robot's environment. Various methods have been developed to learn continuous occupancy maps and have successfull…
SVM predicts regional rainfall with varying accuracy, best in central US.
problem Regional rainfall prediction for social and economic impact planning.
method Support Vector Machine (SVM) applied to sequences of daily rainfall maps.
result SVM predictions for central region outperform untrained classifier.
We investigate the multi-step prediction of the drivable space, represented by Occupancy Grid Maps (OGMs), for autonomous vehicles. Our motivation is that accurate multi-step prediction of the drivable space can efficiently improve path planning and navigation resulting in safe, comfortable and optimum paths in autonom…
Grid Moviesmath.GT
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…
New method models PDEs from noisy, limited data.
problem Modeling PDEs with incomplete, noisy data.
method Learned linear transformation of spatial grid points, followed by dynamics learning in a reduced basis, then back transformation.
result Rapid high-resolution simulations with smaller training data sets.
Modeling regime shifts in co-evolving time series with interactions and time-dependency.
problem Discovering and modeling regime shifts in multiple time series with relationships and time-dependent behaviors.
method Modeling interactions and time-dependency in co-evolving time series using a mapping grid and dynamic network representation for regime identification and time-dependent Cox regression for regime transition probabilities.
result A principled approach for modeling interactions and time-dependency in co-evolving time series.
We study the Ozsváth-Szabó-Thurston transverse invariant in combinatorial link Floer homology for certain transverse cables of transverse link in . Transverse cables are constructed from the grid diagram of . The main result is if and only…
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…
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 under certain conditions.
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.
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.
In recent scene recognition research images or large image regions are often represented as disorganized "bags" of features which can then be analyzed using models originally developed to capture co-variation of word counts in text. However, image feature counts are likely to be constrained in different ways than word …
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.
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…
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.
A novel method visualizes higher-dimensional spaces using hyperbolic geometry.
problem Challenges in visualizing higher-dimensional spaces.
method Interactive visualization of higher-dimensional grids based on hyperbolic geometry.
result Our method shows the whole higher-dimensional space at once and avoids disadvantages of previous methods.
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.
A linear section of a double vector bundle is a parallel pair of sections which form a vector bundle morphism; examples include the complete lifts of vector fields to tangent bundles and the horizontal lifts arising from a connection in a vector bundle. A grid in a double vector bundle consists of two linear sections, …
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.
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.
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.
Power grids are one of the most important components of infrastructure in today's world. Every nation is dependent on the security and stability of its own power grid to provide electricity to the households and industries. A malfunction of even a small part of a power grid can cause loss of productivity, revenue and i…
In this paper, we develop an online method that leverages machine learning to obtain feasible solutions to the AC optimal power flow (OPF) problem with negligible optimality gaps on extremely fast timescales (e.g., milliseconds), bypassing solving an AC OPF altogether. This is motivated by the fact that as the power gr…
The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…
This paper proposes a multi-grid method for learning energy-based generative ConvNet models of images. For each grid, we learn an energy-based probabilistic model where the energy function is defined by a bottom-up convolutional neural network (ConvNet or CNN). Learning such a model requires generating synthesized exam…