7 research outputs found
Topologically Driven Methods for Construction Of Multi-Edge Type (Multigraph with nodes puncturing) Quasi-Cyclic Low-density Parity-check Codes for Wireless Channel, WDM Long-Haul and Archival Holographic Memory
In this Phd thesis discusses modern methods for constructing MET QC-LDPC
codes with a given error correction ("waterfall, error-floor") and complexity
(parallelism level according circulant size plus scheduler orthogonality of
checks) profiles: 1. weight enumerators optimization, protograph construction
using Density Evolution, MI (P/Exit-chart) and it approximation: Gaussian
Approximation, Reciprocal-channel approximation and etc; 2. Covariance
evolution and it approximation; 3. Lifting methods for QC codes
construction:PEG, Guest-and-Test, Hill-Climbing with girth, EMD, ACE
optimization; 4. Upper and lower bounds on code distance estimation and its
parallel implementation using CPU/GPU; 5. Brouwer-Zimmerman and Number Geometry
code distance estimation methods; 6. Importance Sampling for error-floor
estimation; 7. Length and rate adaption methods for QC codes based on cyclic
group decomposition; 8. Methods for interaction screening which allow to
improve performance (decorrelate variables) under BP and it's approximation. We
proposed several state-of-the-art methods: Simulated Annealing lifting for MET
QC-LDPC codes construction; fast EMD and code distance estimation; floor scale
modular lifting for lenght adaption; fast finite-length covariance evolution
rate penalty from threshold for code construction and it hardware friendly
compression for fast decoder's LLRs unbiasing due SNR's estimation error. We
found topology reason's of efficient of such methods using topology thickening
(homotopy of continuous and discrete curvature) under matched metric space
which allow to generalize this idea to a class of nonlinear codes for Signal
Processing and Machine Learning. Using the proposed algorithms several
generations of WDM Long-Haul error-correction codes were built. It was applied
for "5G eMBB" 3GPP TS38.212 and other applications like Flash storage,
Compressed sensing measurement matrix.Comment: Phd Thesis, 176 pages, in Russian, 62 pictures, 13 tables, 5 appendix
including links to binary and source code
Topology-Aware Exploration of Energy-Based Models Equilibrium: Toric QC-LDPC Codes and Hyperbolic MET QC-LDPC Codes
This paper presents a method for achieving equilibrium in the ISING
Hamiltonian when confronted with unevenly distributed charges on an irregular
grid. Employing (Multi-Edge) QC-LDPC codes and the Boltzmann machine, our
approach involves dimensionally expanding the system, substituting charges with
circulants, and representing distances through circulant shifts. This results
in a systematic mapping of the charge system onto a space, transforming the
irregular grid into a uniform configuration, applicable to Torical and Circular
Hyperboloid Topologies. The paper covers fundamental definitions and notations
related to QC-LDPC Codes, Multi-Edge QC-LDPC codes, and the Boltzmann machine.
It explores the marginalization problem in code on the graph probabilistic
models for evaluating the partition function, encompassing exact and
approximate estimation techniques. Rigorous proof is provided for the
attainability of equilibrium states for the Boltzmann machine under Torical and
Circular Hyperboloid, paving the way for the application of our methodology.
Practical applications of our approach are investigated in Finite Geometry
QC-LDPC Codes, specifically in Material Science. The paper further explores its
effectiveness in the realm of Natural Language Processing Transformer Deep
Neural Networks, examining Generalized Repeat Accumulate Codes,
Spatially-Coupled and Cage-Graph QC-LDPC Codes. The versatile and impactful
nature of our topology-aware hardware-efficient quasi-cycle codes equilibrium
method is showcased across diverse scientific domains without the use of
specific section delineations.Comment: 16 pages, 29 figures. arXiv admin note: text overlap with
arXiv:2307.1577
Cyclic Group Projection for Enumerating Quasi-Cyclic Codes Trapping Sets
This paper introduces a novel approach to enumerate and assess Trapping sets
in quasi-cyclic codes, those with circulant sizes that are non-prime numbers.
Leveraging the quasi-cyclic properties, the method employs a tabular technique
to streamline the importance sampling step for estimating the pseudo-codeword
weight of Trapping sets. The presented methodology draws on the mathematical
framework established in the provided theorem, which elucidates the behavior of
projection and lifting transformations on pseudo-codewordsComment: 7 pages, 3 table
Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning
The paper introduces the application of information geometry to describe the
ground states of Ising models by utilizing parity-check matrices of cyclic and
quasi-cyclic codes on toric and spherical topologies. The approach establishes
a connection between machine learning and error-correcting coding. This
proposed approach has implications for the development of new embedding methods
based on trapping sets. Statistical physics and number geometry applied for
optimize error-correcting codes, leading to these embedding and sparse
factorization methods. The paper establishes a direct connection between DNN
architecture and error-correcting coding by demonstrating how state-of-the-art
architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range
arena can be equivalent to of block and convolutional LDPC codes (Cage-graph,
Repeat Accumulate). QC codes correspond to certain types of chemical elements,
with the carbon element being represented by the mixed automorphism
Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and
the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix
are elaborated upon in detail. The Quantum Approximate Optimization Algorithm
(QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous
to the back-propagation loss function landscape in training DNNs. This
similarity creates a comparable problem with TS pseudo-codeword, resembling the
belief propagation method. Additionally, the layer depth in QAOA correlates to
the number of decoding belief propagation iterations in the Wiberg decoding
tree. Overall, this work has the potential to advance multiple fields, from
Information Theory, DNN architecture design (sparse and structured prior graph
topology), efficient hardware design for Quantum and Classical DPU/TPU (graph,
quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text
overlap with arXiv:2109.08184 by other author