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Bayesian Nonlinear Tensor Regression with Functional Fused Elastic Net Prior
Methodology · Submitted 16 Feb 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The great flexibility and interpretability of the proposed method against the alternatives are demonstrated by a simulation study and an analysis on facial feature data.
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The abstract, sentence by sentence
Tensor regression methods have been widely used to predict a scalar response from covariates in the form of a multiway array. In many applications, the regions of tensor covariates used for prediction are often spatially connected with unknown shapes and discontinuous jumps on the boundaries. Moreover, the relationship between the response and the tensor covariates can be nonlinear. In this article, we develop a nonlinear Bayesian tensor additive regression model to accommodate such spatial structure. A functional fused elastic net prior is proposed over the additive component functions to comprehensively model the nonlinearity and spatial smoothness, detect the discontinuous jumps, and simultaneously identify the active regions. The great flexibility and interpretability of the proposed method against the alternatives are demonstrated by a simulation study and an analysis on facial feature data.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Tensor regression methods have been widely used to predict a scalar response from covariates in the form of a multiway array. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | In many applications, the regions of tensor covariates used for prediction are often spatially connected with unknown shapes and discontinuous jumps on the boundaries. | background 0.96 | 0.08 | 0.02 | asserted 0.90 |
| s2 | Moreover, the relationship between the response and the tensor covariates can be nonlinear. | background 0.93 | 0.07 | 0.02 | asserted 0.74 |
| s3 | In this article, we develop a nonlinear Bayesian tensor additive regression model to accommodate such spatial structure. | contribution 0.92 | 0.94 | 0.12 | asserted 1.00 |
| s4 | A functional fused elastic net prior is proposed over the additive component functions to comprehensively model the nonlinearity and spatial smoothness, detect the discontinuous jumps, and simultaneously identify the active regions. | method 0.84 | 0.93 | 0.21 | asserted 0.99 |
| s5 | The great flexibility and interpretability of the proposed method against the alternatives are demonstrated by a simulation study and an analysis on facial feature data. | result 0.76 | 0.91 | 0.52 | asserted 1.00 |
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The paper
Original abstract
Tensor regression methods have been widely used to predict a scalar response from covariates in the form of a multiway array. In many applications, the regions of tensor covariates used for prediction are often spatially connected with unknown shapes and discontinuous jumps on the boundaries. Moreover, the relationship between the response and the tensor covariates can be nonlinear. In this article, we develop a nonlinear Bayesian tensor additive regression model to accommodate such spatial structure. A functional fused elastic net prior is proposed over the additive component functions to comprehensively model the nonlinearity and spatial smoothness, detect the discontinuous jumps, and simultaneously identify the active regions. The great flexibility and interpretability of the proposed method against the alternatives are demonstrated by a simulation study and an analysis on facial feature data.
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What it cites · 3 sentences without a confident label
Uses the method or tool of 2 papers
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For $ρ_$, we specify the grid ${0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 5}$ for $ρ_$ following the suggestion of [ref] .
How Likely are Ride-share Drivers to Earn a Living Wage? Large-scale Spatio-temporal Density Smoothing with the Graph-fused Elastic Net · Validation Method · 0.81
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On the other hand, to compute the degrees of $δ$ and $σ$ when they go to $0$, we apply [ref] 's formula for $n$ dimensional square matrices $,$: [equation] where $$ is an invertible square matrix, and $p_1(·), ⋯, p_n-1(·)$ are the sums of all principal minors of order $2,⋯, n-1$, respectively.
Notes on the combinatorial fundamentals of algebra · The Order of Normalizing Term of the Prior · 0.93
Cites as background 2 papers
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Nonparametric additive models have recently been extended to tensor covariates with elastic net [ref] and the group lasso penalty [ref] .
Broadcasted Nonparametric Tensor Regression · Introduction · 0.81
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With a rank-$1$ CP decomposition $= _1∘⋯∘_D$ where $∘$ denotes the outer product and $_d$ is a $P_d$-dimensional vector, [ref] and [ref] considered a regression model $f() = ∑_r=1^R∏_d=1^D f_r^(d)(_d)$ with a Gaussian process prior over each $f_r^(d)$, $d=1,…, D$.
Learning Tensors in Reproducing Kernel Hilbert Spaces with Multilinear Spectral Penalties · Introduction · 0.84
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