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Concept in linear algebra
Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form
Overcompleteness
Subset of artificial intelligence
M Elad, and A Bruckstein. 2006. "K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation Archived 2018-11-23 at the Wayback
Machine_learning
Integral transform
transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter
Continuous_wavelet_transform
Concept in statistics
provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in
Quasiprobability_distribution
formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm Matrix Matrix addition Matrix multiplication Basis
Outline_of_linear_algebra
Message encoded with more bits than needed
Data compression Hartley function Negentropy Source coding theorem Overcompleteness Here it is assumed A X {\displaystyle {\mathcal {A}}_{X}} are the sets
Redundancy (information theory)
Redundancy_(information_theory)
Vector quantization algorithm minimizing the sum of squared deviations
Michael; Bruckstein, Alfred (2006). "K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation" (PDF). IEEE Transactions on Signal
K-means_clustering
Specific quantum state of a quantum harmonic oscillator
states, expressed as eigenvectors of the lowering operator and forming an overcomplete family, were introduced in the early papers of John R. Klauder, e.g.
Coherent_state
Representation learning method
denoising and classification, and video and audio processing. Sparsity and overcomplete dictionaries have immense applications in image compression, image fusion
Sparse_dictionary_learning
Function for integral Fourier-like transform
respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space, for the Hilbert space of square-integrable
Wavelet
Role of coherent states
\alpha \vert \alpha \rangle =1} . These generalized coherent states are overcomplete in the Fock space and satisfy a resolution of the identity ∫ D | α ⟩
Coherent states in mathematical physics
Coherent_states_in_mathematical_physics
Branch of neuroscience
Olshausen, Bruno A.; Field, David J. (1997-12-01). "Sparse coding with an overcomplete basis set: A strategy employed by V1?". Vision Research. 37 (23): 3311–3325
Computational_neuroscience
Parameter-free superresolution algorithm
adjacent element in the overcomplete dictionary A {\displaystyle {\bf {A}}} , therefore, the difficulty of choosing the optimum overcomplete dictionary arises
SAMV_(algorithm)
Vector satisfying some of the criteria of an eigenvector
formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm Matrices Matrix Matrix addition Matrix multiplication
Generalized_eigenvector
Neural network that learns efficient data encoding in an unsupervised manner
code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or
Autoencoder
Value in matrix theory
representation from a collection with extra building blocks, known as an overcomplete dictionary. Joel Tropp extended this idea with the Babel function, which
Mutual coherence (linear algebra)
Mutual_coherence_(linear_algebra)
Quantum mechanical model
eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality. The coherent states
Quantum_harmonic_oscillator
Neural network coding model
dictionary D {\textstyle \mathbf {D} } . This implies learning large, highly overcomplete representations, which is extremely expensive. Assuming such a burden
Convolutional_sparse_coding
Monochrome light beam whose amplitude envelope is a Gaussian function
Laguerre–Gaussian modes. The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field
Gaussian_beam
Approach in generative models
authors defined a generalisation of independent components analysis to the overcomplete setting using EBMs. Other early work on EBMs proposed models that represented
Energy-based_model
Method by which information is represented in the brain
larger than the dimensionality of the input set, the coding is overcomplete. Overcomplete codings smoothly interpolate between input vectors and are robust
Neural_coding
Israeli computer scientist, a professor of Computer Science (born 1963)
; Bruckstein, A.M. (2006), "The K-SVD: An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation" (PDF), IEEE Transactions on Signal
Michael_Elad
Beninese-American mathematician
Kasso A. (2016). Finite Frame Theory: A Complete Introduction to Overcompleteness. Proceedings of Symposia in Applied Mathematics. ISBN 9781470420192
Kasso_Okoudjou
Set of learning techniques in machine learning
representation from unlabeled input data. Sparse coding can be applied to learn overcomplete dictionaries, where the number of dictionary elements is larger than
Feature_learning
Genetic mutation
complete because a part of the sequence of the chromosome is missing, or overcomplete, because some parts of the sequence are duplicated. The result is that
1q21.1_copy_number_variations
Signal processing computational method
dimensionality of the observed vectors, n > m {\displaystyle n>m} , the task is overcomplete but is still solvable with the pseudo inverse. With the added assumption
Independent component analysis
Independent_component_analysis
Vector function in optics
Bando, Y., Raskar, R. (2013). "Compressive Light Field Photography using Overcomplete Dictionaries and Optimized Projections" Archived 2021-11-28 at the Wayback
Light_field
Similar to the basis of a vector space, but not necessarily linearly independent
{v} \right\|^{2},\quad \forall \mathbf {v} \in V.} A frame is called overcomplete (or redundant) if it is not a Riesz basis for the vector space. The redundancy
Frame_(linear_algebra)
Multidimensional data algorithm
; Bruckstein, A.M. (2006). "The K-SVD: An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation". IEEE Transactions on Signal
Matching_pursuit
American neuroscientist and professor
Gabor filters appearing in the V1 cortex performs sparse coding with overcomplete basis set, such that it is optimal for images occurring in the natural
Bruno_Olshausen
Biological theory of intelligence
Olshausen, Bruno A.; Field, David J. (1997). "Sparse coding with an overcomplete basis set: A strategy employed by V1?". Vision Research. 37 (23): 3311–3325
Hierarchical_temporal_memory
Theoretical model of sensory neuroscience
PMID 13167245. Olshausen, B. A.; Field, D.J. (1997). "Sparse coding with an overcomplete basis set: A strategy employed by V1?". Vision Research. 37 (23): 3311–3325
Efficient_coding_hypothesis
Gauge field loop operator
the gauge connection. Formally the set of all Wilson loops forms an overcomplete basis of solutions to the Gauss' law constraint. The set of all Wilson
Wilson_loop
Quantization method for constrained Hamiltonian systems with second-class constraints
constraint Second class constraints Lagrangian Symplectic structure Overcompleteness Dirac, P. A. M. (1950). "Generalized Hamiltonian dynamics". Canadian
Dirac_bracket
Group of associated animal fossils found together in a given stratum
11.008. ISSN 1040-6182. Kowalewski, Michał (1996). "Time-Averaging, Overcompleteness, and the Geological Record". The Journal of Geology. 104 (3): 317–326
Faunal_assemblage
Spectral density estimation technique
recent years, a dictionary based on modulated DPSS was proposed as an overcomplete alternative to DPSS. Not limited to time series, the multitaper method
Multitaper
Concept in mathematics
D.L., Elad, M. and Templyakov, V. (2006). "Stable recovery of sparse overcomplete representations in the presence of noise" (PDF). IEEE Transactions on
Sparse_approximation
The disadvantage of the steerable pyramid decomposition is that it is overcomplete. This means that more filters than truly necessary are used to describe
Log_Gabor_filter
Romanian scientist and emeritus professor
Elad for their paper titled "The K-SVD: An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation". Doctor Honoris Causa of Agora
Alfred_Marcel_Bruckstein
Dictionary learning algorithm
Michael Elad; Alfred Bruckstein (2006), "K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation" (PDF), IEEE Transactions on Signal
K-SVD
Subfield of cognitive science
3327–3338. Olshausen, B. A., & Field, D. J. (1997). Sparse coding with an overcomplete basis set: A strategy by V1? Vision Research, 37(23), 3311–3325. Sigman
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