scholarly journals Statistical Learning in Tree-Based Tensor Format

2018 ◽  
Author(s):  
Erwan Grelier ◽  
Mathilde Chevreuil ◽  
Anthony Nouy
Keyword(s):  
Statistical Learning ◽  
Learning Algorithm ◽  
Projection Pursuit ◽  
Singular Values ◽  
Tensor Methods ◽  
Tensor Networks ◽  
Dimension Partition ◽  
Tensor Approximation ◽  
P Tensor ◽  
New Variables

Tensor methods are widely used tools for the approximation of high dimensional functions. Such problems are encountered in uncertainty quantification and statistical learning, where the high dimensionality imposes to use specific techniques, such as rank-structured approximations [1]. In this work, we introduce a statistical learning algorithm for the approximation in tree-based tensor format, which are tensor networks whose graphs are dimension partition trees. This tensor format includes the Tucker format, the Tensor-Train format, as well as the more general Hierarchical tensor formats [4]. It can be interpreted as a deep neural network with a particular architecture [2]. The proposed algorithm uses random evaluations of a function to provide a tree-based tensor approximation, with adaptation of the tree-based rank by using a heuristic criterion based on the higher-order singular values to select the ranks to increase, and of the approximation spaces of the leaves of the tree. We then present a learning algorithm for the approximation under the form u(x) ≈ v(z_1,...,z_m) where v is a tensor in tree-based format and the z_i = g_i(x), 1 ≤ i ≤ m, are new variables. A strategy based on the projection pursuit regression [3] is proposed to compute the mappings g_i and increase the effective dimension m. The methods are illustrated on different examples to show their efficiency and adaptability as well as the power of representation of the tree-based tensor format, possibly combined with changes of variables.

scholarly journals GENERALIZED CELLULAR NEURAL NETWORKS (GCNNs) CONSTRUCTED USING PARTICLE SWARM OPTIMIZATION FOR SPATIO-TEMPORAL EVOLUTIONARY PATTERN IDENTIFICATION

2008 ◽  
Vol 18 (12) ◽  
pp. 3611-3624 ◽  
Author(s):  
H. L. WEI ◽  
S. A. BILLINGS
Keyword(s):  
Neural Networks ◽  
Particle Swarm Optimization ◽  
Orthogonal Projection ◽  
Learning Algorithm ◽  
Projection Pursuit ◽  
Particle Swarm ◽  
Cellular Neural Networks ◽  
Modeling Framework ◽  
Swarm Optimization ◽  
Spatio Temporal

Particle swarm optimization (PSO) is introduced to implement a new constructive learning algorithm for training generalized cellular neural networks (GCNNs) for the identification of spatio-temporal evolutionary (STE) systems. The basic idea of the new PSO-based learning algorithm is to successively approximate the desired signal by progressively pursuing relevant orthogonal projections. This new algorithm will thus be referred to as the orthogonal projection pursuit (OPP) algorithm, which is in mechanism similar to the conventional projection pursuit approach. A novel two-stage hybrid training scheme is proposed for constructing a parsimonious GCNN model. In the first stage, the orthogonal projection pursuit algorithm is applied to adaptively and successively augment the network, where adjustable parameters of the associated units are optimized using a particle swarm optimizer. The resultant network model produced at the first stage may be redundant. In the second stage, a forward orthogonal regression (FOR) algorithm, aided by mutual information estimation, is applied to refine and improve the initially trained network. The effectiveness and performance of the proposed method is validated by applying the new modeling framework to a spatio-temporal evolutionary system identification problem.

scholarly journals Statistical Learning Algorithm for in situ and invasive breast carcinoma segmentation

2013 ◽  
Vol 37 (4) ◽  
pp. 281-292 ◽  
Author(s):  
Jagadeesan Jayender ◽  
Eva Gombos ◽  
Sona Chikarmane ◽  
Donnette Dabydeen ◽  
Ferenc A. Jolesz ◽  
...  
Keyword(s):  
Breast Carcinoma ◽  
Statistical Learning ◽  
Learning Algorithm ◽  
Invasive Breast Carcinoma

scholarly journals The Novel Quantitative Technique for Assessment of Gait Symmetry Using Advanced Statistical Learning Algorithm

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Jianning Wu ◽  
Bin Wu
Keyword(s):  
Statistical Learning ◽  
Learning Algorithm ◽  
Lower Limbs ◽  
Classification Method ◽  
Support Vector ◽  
Accurate Identification ◽  
Index Method ◽  
Gait Symmetry ◽  
Significant Difference ◽  
Gait Asymmetry

The accurate identification of gait asymmetry is very beneficial to the assessment of at-risk gait in the clinical applications. This paper investigated the application of classification method based on statistical learning algorithm to quantify gait symmetry based on the assumption that the degree of intrinsic change in dynamical system of gait is associated with the different statistical distributions between gait variables from left-right side of lower limbs; that is, the discrimination of small difference of similarity between lower limbs is considered the reorganization of their different probability distribution. The kinetic gait data of 60 participants were recorded using a strain gauge force platform during normal walking. The classification method is designed based on advanced statistical learning algorithm such as support vector machine algorithm for binary classification and is adopted to quantitatively evaluate gait symmetry. The experiment results showed that the proposed method could capture more intrinsic dynamic information hidden in gait variables and recognize the right-left gait patterns with superior generalization performance. Moreover, our proposed techniques could identify the small significant difference between lower limbs when compared to the traditional symmetry index method for gait. The proposed algorithm would become an effective tool for early identification of the elderly gait asymmetry in the clinical diagnosis.

Fast Statistical Learning Algorithm for Feature Generation

2007 ◽  
pp. 91-97 ◽  
Author(s):  
Minoru Fukumi ◽  
Stephen Karungaru ◽  
Satoru Tsuge ◽  
Miyoko Nakano ◽  
Takuya Akashi ◽  
...  
Keyword(s):  
Statistical Learning ◽  
Learning Algorithm ◽  
Feature Generation

Characterizing and detecting climate signals in observations and models using statistical learning

2020 ◽  
Author(s):  
Sebastian Sippel ◽  
Nicolai Meinshausen ◽  
Erich Fischer ◽  
Eniko Szekely ◽  
Reto Knutti
Keyword(s):  
Statistical Learning ◽  
Spatial Patterns ◽  
Climate Model ◽  
Learning Algorithm ◽  
Global Climate ◽  
Climatic Changes ◽  
Learning Techniques ◽  
Statistical Learning Techniques ◽  
Climate Signals ◽  
Global Mean

<p>Internal atmospheric variability fundamentally limits short- and medium-term climate predictability and obscures evidence of climatic changes on regional scales. We discuss the suitability of incorporating statistical learning techniques to detect global climate signals from spatial patterns.</p><p>Our detection approach uses climate model simulations and a statistical learning algorithm to encapsulate the relationship between spatial patterns of daily temperature and humidity, and key climate change metrics such as annual global mean temperature or Earth’s energy imbalance. Observations are then projected onto this relationship to detect climatic changes. We show that fingerprints of changes in climate can be assessed and detected in the observed global climate record at time steps such as months or days by comparison against a historical baseline from CMIP5 simulations or reanalyses. Detection can be achieved also when ignoring the long-term global mean warming trend.</p><p>We further discuss how these approaches could be extended by using statistical techniques that would work well under variations of specific external forcings, e.g. solar or volcanic forcing, to predict only variations in a specific external forcing. Overall, we conclude that statistical learning techniques that characterize multivariate signals from high-dimensional climate data are a useful tool for the detection of climate signals at regional and global scales.</p>

scholarly journals An orthogonal equivalence theorem for third order tensors

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Liqun Qi ◽  
Chen Ling ◽  
Jinjie Liu ◽  
Chen Ouyang
Keyword(s):  
Singular Values ◽  
Equivalence Theorem ◽  
Low Rank ◽  
Third Order ◽  
Order Tensor ◽  
Rank Tensor ◽  
Tensor Recovery ◽  
Rank One ◽  
Tensor Approximation ◽  
Value Decomposition

<p style='text-indent:20px;'>In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal tensors. We define singular values and T-rank of that third order tensor based upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular values and T-singular values of a third order tensor are invariant when it is taking T-product with some orthogonal tensors. Some properties of singular values, T-rank and best T-rank one approximation are discussed.</p>

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