Neural Lattice Point Enumeration for Lower-Complexity Counting
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Solution Overview
Problem
Existing techniques for solving lattice-related problems, such as lattice points enumeration and closest vector problems, require high computational complexity and are challenging to implement in practical systems due to their super exponential time and memory requirements, especially as the lattice dimension increases.
Innovation Solution
A lattice prediction device and method using a machine learning algorithm, specifically a multilayer deep neural network, to predict the number of lattice points within a bounded region by processing input data derived from the radius value and components of the lattice generator matrix, reducing computational complexity through QR decomposition and optimization algorithms.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If existing techniques (lattice enumeration, random sampling reduction, lattice sieving, Voronoi cell computation, discrete Gaussian sampling) are used to solve lattice points enumeration problem, then the problem can be solved, but the computational complexity becomes super exponential or high polynomial scaling
Solution Approach 1:
The patent replaces traditional mechanical/computational lattice enumeration methods with a deep neural network-based system. The neural network model (including encoder and decoder components) processes lattice parameters and predicts the number of lattice points within a bounded region, substituting complex algorithmic computations with learned patterns from training data. This substitution dramatically reduces computational complexity from super-exponential to much lower complexity levels.
Solution Approach 2:
The patent transforms the problem by changing the input parameters and representation. Instead of working with the original lattice basis directly, the system uses transformed parameters through QR decomposition and processes them through neural network layers. The model takes lattice dimension, basis vectors, and bounding region parameters as inputs, and outputs the predicted number of lattice points, effectively changing the computational landscape to be more manageable.
2Measurement precision
If traditional lattice problem solving algorithms are implemented, then exact solutions can be obtained, but the algorithms become challenging for practical implementations due to high computational requirements
Solution Approach 1:
The patent replaces traditional algorithmic implementations with a neural network-based system that processes lattice problems through learned representations. The neural network model, trained on various lattice configurations, provides predictions that are both accurate and computationally feasible for practical implementations, eliminating the need for complex exact algorithms.
Solution Approach 2:
The patent creates a computational model that copies and generalizes patterns from training data. The neural network learns from example lattice configurations and applies these patterns to new problems, providing accurate solutions without requiring re-implementation of complex algorithms for each specific case. This copying approach enables practical implementation through a single trained model.
Data Source
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AI summary
A lattice prediction device (200) for predicting a number of lattice points falling inside a bounded region in a given vector space. The bounded region is defined by a radius value, a lattice point representing a digital signal in a lattice constructed over the vector space. The lattice is defined by a lattice generator matrix comprising components. The lattice prediction device (200) comprises a computation unit (201) configured to determine a predicted number of lattice points by applying a machine learning algorithm to input data derived from the radius value and the components of lattice generator matrix.