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Partial Modal Tensor Regression

Jintao Han, Li Zhang, Xuejun Ma

arXiv:2610.34008v1stat.MLcs.LG

TL;DR

Tensor regression must preserve multiway structure while addressing high dimensionality, and methods for conditional-modal modeling with tensor predictors remain less explored. PMTR uses partial modal tensor covariance to guide dimension reduction and fits a modal model in the reduced space. Simulations and a Beijing air-quality analysis report stable recovery and interpretable environmental association patterns.

  • Problem

    Tensor regression methods have chiefly targeted conditional means or quantiles, leaving conditional-modal regression with tensor predictors less explored despite its relevance to asymmetric, heavy-tailed, or contaminated responses.

    Source: §1 · pp. 2–3

  • Method

    PMTR constructs partial modal tensor covariance to guide mode-wise dimension reduction while preserving tensor structure, then models the conditional modal in the reduced space.

    Source: §1 · p. 3, §7 · p. 22

  • Results

    Simulations report stable coefficient-structure recovery with accuracy improving as sample size increases, and the Beijing air-quality analysis identifies interpretable environmental association patterns that vary over the day.

    Source: §7 · p. 23

  • Takeaways & Limitations

    PMTR provides a structure-preserving approach to conditional-modal analysis of high-dimensional tensor predictors.

    Source: §7 · p. 22

    The study focuses primarily on independent observations; sparse regularization and more systematic uncertainty quantification and inference remain future directions.

    Source: §7 · p. 23

Abstract

from arXiv · show

With the rapid development of modern scientific research, tensor data have become increasingly prevalent across a wide range of fields. How to effectively characterize the relationship between tensor predictors and a response while preserving the intrinsic structure of tensor data has therefore become an important problem in high-dimensional statistical analysis. Existing tensor regression methods mainly focus on conditional means or conditional quantiles. When the response distribution is asymmetric, heavy-tailed, or contaminated by outlying observations, the conditional modal provides a robust and interpretable alternative characterization. In this paper, we propose a partial modal tensor regression (PMTR) method. The proposed method constructs the partial modal tensor covariance, which incorporates information on the conditional modal to guide tensor dimension reduction, and subsequently establishes a modal tensor regression model in the reduced tensor space. This strategy alleviates the dimensionality burden caused by direct vectorization while preserving the intrinsic structural information of tensor predictors. Under suitable regularity conditions, we establish the consistency and asymptotic properties of the proposed estimators and employ a data-driven dimension-selection procedure to determine the effective dimension along each tensor mode.

1 Introduction

Tensor predictors preserve multiway structure, but direct vectorization can inflate the parameter space. Existing tensor regression chiefly targets means or quantiles, leaving conditional-modal regression with tensor covariates less explored.

  • Direct vectorization ignores tensor structure and makes the regression parameter dimension grow multiplicatively with the mode dimensions.Source: §1 · p. 2
  • Most tensor regression methods target conditional means or quantiles, while methods for conditional-modal regression with tensor predictors remain less explored.Source: §1 · pp. 2–3
  • Conditional modes characterize the most probable response and can be informative for asymmetric, heavy-tailed, or contaminated response distributions.Source: §1 · p. 3
  • PMTR combines conditional-modal modeling with response-guided tensor dimension reduction and reports interpretable hour-specific environmental associations in Beijing air-quality data.Source: §1 · p. 3

2 Partial Modal Tensor Regression

The section defines a conditional-modal regression model for tensor and vector covariates, then motivates dimension reduction because the tensor coefficient parameter count can overwhelm the sample size.

  • The model treats the conditional modal of Y given tensor X and vector Z as a linear function of the covariates.Source: §2 · p. 4
  • The coefficient tensor B represents X's contribution, while the intercept and vector coefficient are grouped as the marginal linear term in Z.Source: §2 · p. 4
  • The modal regression estimator uses a kernel-based objective, with bandwidth selection following Yao and Li (2014).Source: §2 · p. 4
  • When the product of tensor mode dimensions is much larger than sample size, direct estimation risks non-identifiability or overfitting and may be infeasible.Source: §2 · p. 5

3 Modal Tensor PLS Algorithm

The algorithm uses partial modal tensor covariance to guide Tucker-based, mode-wise dimension reduction, then fits modal regression in the reduced space. A ridge-type eigenvalue ratio selects each mode's dimension.

  • Tucker decomposition guided by the PLS criterion provides mode-wise dimension reduction, using a separable Kronecker covariance structure for tensor X.Source: §3 · p. 5
  • Partial modal tensor covariance is proposed to reduce the tensor predictor's dimension while accounting for the response and adjusting for the marginal modal effect of Z.Source: §3 · p. 5
  • Under the stated conditions, this covariance has a residualized pseudo-linear representation of the coefficient tensor with a smoothing remainder, and it is zero if and only if B0 is zero under the specified zero-score and nonsingularity conditions.Source: §3 · p. 6
  • The ridge-type eigenvalue ratio selects each mode's dimension, reducing computational burden relative to cross-validation and avoiding the conventional eigenvalue ratio's small-denominator instability.Source: §3 · p. 8

4 Theoretical Properties

The section establishes asymptotic properties for the reduced-space estimator under regularity conditions and justifies data-driven selection of the mode-wise dimensions through the RER criterion.

  • With correct mode-wise reduction dimensions and bandwidths of order n^-1/7, Theorem 1 establishes a local maximizer with probability tending to one and characterizes the estimator's asymptotic behavior.Source: §4 · p. 11
  • The theorem's asymptotic results assume the true mode dimensions are correctly specified, although those dimensions are generally unknown in practice.Source: §4 · pp. 11–12
  • The RER criterion consistently selects each mode's dimension: its ratio diverges at the true dimension while competing ratios remain stochastically bounded.Source: §4 · p. 12

5 Simulation Studies

The simulations assess PMTR’s coefficient-tensor recovery across structural patterns, sample sizes, error distributions, and noise levels. Recovery improves with sample size, while higher rank and more noise make estimation harder.

5 Simulation Studies

  • Each simulation setting is repeated 100 times; coefficient recovery is assessed using average estimated tensors and absolute Frobenius estimation errors, with smaller errors indicating greater accuracy.The simulation varies coefficient structure, sample size, error distribution, and noise level; the predictor and true tensor coefficient are 50 × 50.Source: §5 · pp. 12–13
  • In small samples, higher-rank coefficient structures pose greater estimation challenges and may show spurious signals; at n = 200 or 400, recovery improves, with Frobenius errors approximately 5 or lower in most settings at n = 200.Source: §5 · p. 16
  • Across Gaussian, t3, log-normal, and contaminated normal errors, recovery patterns are broadly similar, although t3 errors are slightly larger than the other three distributions.The authors describe the distribution effect as relatively modest and overall error levels as comparable.Source: Figure 4 · p. 17, §5 · p. 18
  • As noise increases, spurious signals and absolute Frobenius errors rise gradually, yet the main coefficient structure remains identifiable even at 200% noise.The triangle-shaped coefficient tensor has rank 9, and these estimates use n = 200.Source: Figure 3 · p. 16, Figure 4 · p. 17, §5 · p. 18
  • PMTR recovers main coefficient structure across the tested settings; recovery improves with sample size, while higher rank and noise increase estimation difficulty.The authors report stable performance across Gaussian, t3, log-normal, and contaminated normal errors, with main structural features still identifiable under substantial noise.Source: §5 · pp. 18–19

6 An Application to the Beijing Air-Quality Data

The application models daily average PM2.5 using each day’s 24-by-8 hour-by-variable predictor matrix and seasonal covariates. PMTR selects an 8-by-8 representation and identifies stable, time-varying associations, especially for NO2, O3, TEMP, and PRES.

6 An Application to the Beijing Air-Quality Data

  • The selected-day predictor is a 24 × 8 matrix of hourly measurements for eight pollutants and meteorological variables, with daily average PM2.5 as the response.The study uses Wanliu monitoring-station data from March 2013 to February 2017.Source: §6 · p. 19
  • The daily average PM2.5 distribution is right-skewed, with most observations at lower concentrations and a pronounced upper tail.The sample mean exceeds the median, motivating conditional-mode modeling rather than focusing solely on the conditional mean.Source: §6 · p. 20, Figure 5 · p. 20
  • The RER criterion reduces the 24 × 8 predictor to an effective 8 × 8 representation, reducing the hourly dimension while retaining all eight environmental-variable dimensions.The selected mode dimensions are 8 and 8.Source: §6 · p. 20
  • NO2, O3, TEMP, and PRES show the most pronounced stable coefficient patterns across thresholds, with association strength and direction varying over the day.NO2 and O3 have similar temporal patterns; TEMP and PRES also show changing positive and negative coefficients across hours.Source: §6 · pp. 21–22, Figure 6 · p. 21

7 Conclusion

PMTR models the conditional modal of a response using mode-wise dimension reduction that preserves tensor structure. The paper establishes estimator consistency and asymptotic properties, reports stable simulation recovery and interpretable Beijing air-quality associations, and identifies independent observations as its primary scope.

7 Conclusion

  • PMTR combines tensor dimension reduction and modal regression to characterize conditional-modal relationships while preserving multiway predictor structure.The framework avoids dimensional inflation and structural information loss from direct vectorization.Source: §7 · p. 22
  • The proposed estimators have established consistency and asymptotic properties under suitable regularity conditions.Source: §7 · p. 22
  • Simulations show stable coefficient-structure recovery, with accuracy improving as sample size increases and satisfactory performance across error distributions and noise levels.The Beijing analysis also identifies interpretable environmental association patterns that vary over the day.Source: §7 · p. 23
  • The study focuses primarily on PMTR under independent observations; sparse regularization and more systematic uncertainty quantification and inference remain future directions.Source: §7 · p. 23

A.1 Proof of Lemma 1

The proof links the partial modal tensor covariance to the tensor coefficient under stated conditions. It also uses a first-order condition for the population marginal objective and controls a smoothing expansion for asymmetric conditional errors.

A.1 Proof of Lemma 1

  • The proof uses the first-order condition for Y|Z at an interior maximizer of the population marginal objective.Source: §A.1 · p. 26
  • When the conditional residual expectation is zero and V is nonsingular, the proof derives an exact identity and concludes that zero partial modal tensor covariance is equivalent to B0 being zero.Source: §A.1 · p. 27
  • For asymmetric conditional errors, integration by parts and a Taylor expansion establish the smoothing-term order under dominated-expansion and integrability conditions.Source: §A.1 · p. 28

A.2 Proof of Lemma 2

The proof organizes the estimator argument into empirical, plug-in, and Taylor-remainder bounds. It uses independence and stated assumptions to combine these controls and complete the proof.

A.2 Proof of Lemma 2

  • The proof first controls the oracle empirical term, using independence of observations and Assumption (A5) in its bound.Source: §A.2 · p. 29
  • Subsequent steps control the first- and second-order plug-in terms, using integration by parts and assumptions to bound the relevant quantities.Source: §A.2 · pp. 29–30
  • The proof controls the Taylor remainder through localization and bounds, then combines the preceding bounds to complete the argument.Source: §A.2 · pp. 31–32

A.3 Proof of Lemma 3

The proof combines smooth-map delta-method arguments, integration by parts, and operator-norm bounds. It also invokes covariance-matrix results for the mode-k empirical quantity.

A.3 Proof of Lemma 3

  • The proof invokes Lemma 2 and Lemma S6 of Sun et al. (2025) for the empirical mode-k covariance matrix, then combines bounds using Kronecker-product operator norms.Source: §A.3 · pp. 33–34
  • The inverse square-root map is continuously differentiable on positive-definite matrices, allowing use of the finite-dimensional delta method under Assumption (A6).Source: §A.3 · p. 33
  • Integration by parts and Assumption (A11) yield a uniformly bounded quantity, which is then used with Assumption (A5).Source: §A.3 · p. 34

A.4 Proof of Lemma 4

These appendixed proofs establish consistency and joint dimension-selection consistency through perturbation, smoothness, and kernel-score bounds. Theorem 1's proof constructs a local maximizer, while Theorem 2's proof establishes joint selection consistency.

A.4 Proof of Lemma 4

  • Lemma 4's proof proceeds by induction to establish consistency of the initial matrix, first direction, and first deflation matrix; positive scalar factors do not change population directions.Source: §A.4 · pp. 34–35

A.5 Proof of Lemma 5

  • Lemma 5's proof controls the estimated design and kernel derivatives, then decomposes the score difference; its density bounds rely on Assumption (A11).Source: §A.5 · pp. 36, 37, 39

A.6 Proof of Theorem 1

  • Theorem 1's proof obtains a local maximizer by bounding the score at order τ_n and making the boundary objective smaller than its value at η_0.The proof sets τ_n = n^-2/7 and uses continuity and an interior condition.Source: §A.6 · pp. 40, 42

A.7 Proof of Theorem 2

  • The proof establishes joint selection consistency; the cited eigenvalue expression gives k_j + O_p(n^-2/7) for j = 1, ..., r_k.Source: §A.7 · pp. 43–44
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