Label Projection Correlation (PCor)

• Label Projection Correlation (PCor) is a normalized statistic that measures the dependence between a multivariate numerical vector and a categorical label, equaling zero if they are independent.
• It aggregates discrepancies in one-dimensional projections of the feature distribution across different labels, avoiding moment conditions and eliminating the need for kernel tuning.
• PCor estimation employs sorted ranking methods and triplewise angle computations, leading to efficient feature screening and robust performance even under heavy-tailed noise.

Searching arXiv for the cited paper and closely related label-correlation / projection work. arXiv search query: (Liu et al., 27 Apr 2025) OR "Measuring Feature-Label Dependence Using Projection Correlation Statistic" OR "Label Projection Correlation" Label Projection Correlation (PCor) is a normalized dependence measure between a numerical variable/vector $X$ and a categorical label $Y$. It is introduced as label projection correlation, with PCor as the abbreviation, while PCov denotes the corresponding unnormalized projection-covariance-type quantity. The construction measures how much the distribution of a numerical feature/vector changes across categorical labels after averaging discrepancies over one-dimensional projections, and it is characterized by the property $\mathrm{PCor}(X,Y)=0 \iff X$ and $Y$ are independent (Liu et al., 27 Apr 2025).

1. Problem setting and motivation

PCor is defined for the heterogeneous setting

• $X \in \mathbb{R}^p$: a numerical random variable/vector,
• $Y \in \mathcal{Y}=\{1,\dots,K\}$: a categorical random label.

The stated goal is to quantify and test dependence between a numerical feature (or feature vector) and a categorical label. The proposed measure is positioned against several existing numerical–categorical dependence statistics. The comparison given in the source text is specific: the MV method is only for 1D numerical variable + categorical label; Gini distance correlation requires moment conditions such as $E|X|<\infty$; and RKHS-based Gini correlation removes some moment restrictions but requires choosing a kernel and tuning kernel parameters (Liu et al., 27 Apr 2025).

Against that background, PCor is designed to satisfy five properties stated in the source material: it works for multivariate $X$, requires no moment condition on $X$, does not require kernel selection or tuning, equals zero iff $X$ and $Y$0 are independent, and is orthogonally invariant. The terminology in the source is also explicit: “projection correlation statistic” is the broader phrase appearing in the title, whereas the method itself is specifically named label projection correlation.

The construction starts from the criterion that $Y$1 and $Y$2 are independent iff, for every projection direction $Y$3 with $Y$4, the scalar projection $Y$5 is independent of $Y$6. Writing $Y$7 for the c.d.f. of $Y$8, $Y$9 for the conditional c.d.f. given $\mathrm{PCor}(X,Y)=0 \iff X$0, and $\mathrm{PCor}(X,Y)=0 \iff X$1, independence is equivalent to

$\mathrm{PCor}(X,Y)=0 \iff X$2

PCor measures violations of this equality by integrating projected distributional discrepancies over labels, directions, and thresholds.

2. Population definition and geometric form

The population construction begins from the discrepancy functional

$\mathrm{PCor}(X,Y)=0 \iff X$3

Using indicator notation, this is rewritten as

$\mathrm{PCor}(X,Y)=0 \iff X$4

A key geometric identity converts the spherical integral into an angle-based expectation over triples of points. This yields the projection-covariance form

$\mathrm{PCor}(X,Y)=0 \iff X$5

with

$\mathrm{PCor}(X,Y)=0 \iff X$6

and

$\mathrm{PCor}(X,Y)=0 \iff X$7

The equivalent variance representation is

$\mathrm{PCor}(X,Y)=0 \iff X$8

This form makes the mechanism transparent: for each projection $\mathrm{PCor}(X,Y)=0 \iff X$9, one compares the class-conditional probabilities

$Y$0

If the projected distribution does not vary with the label, the conditional variance over $Y$1 vanishes.

Normalization is introduced through the upper bound

$Y$2

The normalized label projection correlation is then

$Y$3

The source states that $Y$4 when $Y$5.

For continuous $Y$6, the expression simplifies substantially: $Y$7 hence

$Y$8

In this regime, the dependence is fully determined by the within-class angle term $Y$9.

3. Estimation, computational regimes, and asymptotics

For multivariate data, the sample estimator is defined through triplewise angles

$X \in \mathbb{R}^p$0

The source specifies special conventions for degenerate cases: $X \in \mathbb{R}^p$1 when only one of $X \in \mathbb{R}^p$2, $X \in \mathbb{R}^p$3 is zero, and $X \in \mathbb{R}^p$4 when both are zero. If $X \in \mathbb{R}^p$5 is the set of indices with label $X \in \mathbb{R}^p$6, $X \in \mathbb{R}^p$7, and $X \in \mathbb{R}^p$8, then

$X \in \mathbb{R}^p$9

$Y \in \mathcal{Y}=\{1,\dots,K\}$0

$Y \in \mathcal{Y}=\{1,\dots,K\}$1

and

$Y \in \mathcal{Y}=\{1,\dots,K\}$2

The direct multivariate estimator has computational cost $Y \in \mathcal{Y}=\{1,\dots,K\}$3.

The one-dimensional case has a distinct definition: $Y \in \mathcal{Y}=\{1,\dots,K\}$4 For the transformed variable $Y \in \mathcal{Y}=\{1,\dots,K\}$5, the paper introduces

$Y \in \mathcal{Y}=\{1,\dots,K\}$6

together with reverse-order analogues $Y \in \mathcal{Y}=\{1,\dots,K\}$7, and then states

$Y \in \mathcal{Y}=\{1,\dots,K\}$8

The computational contribution emphasized in the source is that, in one dimension, estimation can be reduced to sorting. If $Y \in \mathcal{Y}=\{1,\dots,K\}$9 are the full-sample ordered values and $E|X|<\infty$0 are the within-class ordered values, then

$E|X|<\infty$1

$E|X|<\infty$2

The total complexity is then $E|X|<\infty$3.

For one-dimensional continuous $E|X|<\infty$4, the paper further gives the rank-based expression

$E|X|<\infty$5

The asymptotic theory is stated in two regimes. Under dependence,

$E|X|<\infty$6

and

$E|X|<\infty$7

where $E|X|<\infty$8 in the multivariate case and $E|X|<\infty$9 in the one-dimensional case. Under independence, the limiting law is degenerate: $X$0

$X$1

The source text states that the paper motivates PCor as a measure to test independence, but does not provide a fully worked-out standalone testing procedure with calibrated critical values.

4. Theoretical properties

The central theoretical results are summarized as Theorem 2 in the source material (Liu et al., 27 Apr 2025). The first property is boundedness: $X$2 This follows from nonnegativity of the variance representation together with the normalization by the derived upper bound.

The second property is the exact characterization of independence: $X$3 This gives PCor the status of an independence measure rather than a linear-association coefficient.

The third property characterizes the extremal case $X$4. The condition stated in the source is that, for every label $X$5, $X$6 is almost surely a constant vector $X$7, and the $X$8 are not all identical. In words, each class collapses to its own deterministic point and different classes correspond to different points.

The fourth property is invariance: $X$9 for any orthogonal matrix $X$0, any nonzero scalar $X$1, and any vector $X$2. Thus PCor is invariant to translation, nonzero scalar rescaling, and orthogonal transformations.

Taken together, these properties distinguish PCor from classical correlations such as Pearson, Spearman, or Kendall in two ways stated or implied by the source material: it is designed for the heterogeneous setting numerical vs. categorical, and it is built as a zero iff independent dependence functional rather than a measure of merely linear or monotone association. This suggests a different intended use: feature screening and independence assessment rather than correlation analysis between two numerical coordinates.

5. Feature selection and empirical evidence

A major application in the source paper is feature screening / feature selection with a categorical response. The procedure described is univariate: for each feature $X$3, compute $X$4, rank features in descending order, and treat features with larger PCor as more strongly associated with the categorical label (Liu et al., 27 Apr 2025).

The simulation studies compare PCor against gCor, gKCor, and Pearson correlation. The evaluation metrics listed in the source are MMS (minimum model size containing all active variables), RSD (standard deviation of MMS over repetitions), $X$5 (probability of including a specific active variable $X$6 in a selected model of size $X$7), and $X$8 (probability of including all active variables).

In Simulation 1, the design uses a categorical label $X$9 with 3 classes, balanced / slightly unbalanced / heavily unbalanced class proportions, 400 groups/features, sample size $X$0, 60 repetitions, two signal-strength cases, and noise from $X$1, $X$2, and $X$3. The reported finding is that PCor outperforms the other three methods in all cases for this grouped feature selection problem. In Simulation 2, a GWAS-style design is used with SNP-type features encoded into 3 categories, a nonlinear response involving five active SNPs, $X$4, $X$5, 100 repetitions, and the same three noise families. The source states that under $X$6, Pearson is relatively strong; under $X$7, PCor performed best; and under $X$8, PCor and gCor were best on $X$9 and $Y$00.

Two real-data applications are also reported. On DrivFace, the data consist of 606 images from 4 drivers, each standardized to $Y$01 pixels, with gaze-direction labels “looking right”, “frontal”, and “looking left”. The procedure uses a 70%/30% train–test split, computes feature-label dependence for each pixel, selects the top $Y$02 pixels, trains a random forest with 100 trees, evaluates test accuracy, and averages over 10 runs. The source states that the high-PCor pixels are more concentrated on the facial region, and that PCor achieved the highest test accuracy for all listed values of $Y$03.

On the Arcene cancer mass-spectrometry data, the task is binary classification, the training and validation sets each contain 100 samples, there are 10,000 features, and many features have heavy-tailed distributions. The same ranking-and-selection protocol is used, again followed by a random forest with 100 trees and averaging over 10 runs. The reported findings are that PCor and gKCor generally outperform methods with stronger moment sensitivity, that PCor consistently exceeds gCor, and that when the number of selected features is small, PCor shows significantly higher accuracy than competitors.

The source also notes a limitation: because the procedure is univariate screening, correlated features may lead to inappropriate selections. A plausible implication is that PCor is strongest as a marginal dependence statistic, while conditional screening would require additional methodology.

6. Relation to projection- and correlation-based label modeling

Although the term Label Projection Correlation (PCor) is explicit only in the 2025 dependence-measure paper, several earlier arXiv works are closely related at the level of projection, label structure, or correlation preservation. Their relationship to PCor is conceptual rather than terminological.

Method	Core mechanism	Relation to PCor
VPCME	variable pairwise constraint projection	label-guided projection, but via sample-pair constraints
CAMEL	sparse directed label collaboration matrix	explicit label-space propagation/projection operator
GLOCAL	latent label representation + learned Laplacians	latent projection plus manifold regularization
LCG	covariance-based label grid + KPCA regularization	parallel correlation modeling and label projection
LESC / gLESC	low-rank self-expression correlation regularization	correlation-preserving label-distribution recovery

VPCME defines Variable Pairwise Constraint projection for Multi-label Ensemble, where labels induce must-link and cannot-link sample pairs through thresholded label overlap, and the projection maximizes $Y$04 under $Y$05. The source explicitly notes that this is projection-based, label-informed, and intended to preserve “correlations between samples and labels,” but that the preserved structure is implemented through pairwise sample constraints induced by labels, not through an explicit PCor formulation over labels (Li et al., 2014).

CAMEL learns a directed sparse label collaboration matrix $Y$06 by reconstructing each label from the other labels, then forms $Y$07 and uses $Y$08 or $Y$09 as the correlated prediction space. The source characterizes this as a learned label correlation projection/propagation model. It is therefore close to a “projection operator in label space,” but it is not a feature-label dependence coefficient of the PCor type (Feng et al., 2019).

GLOCAL combines low-rank latent label decomposition $Y$10, feature-to-latent prediction $Y$11, and global/local label-manifold regularization through learned Laplacians $Y$12. The source stresses that it is not literally a PCor paper, but is strongly related because it unifies latent label projection with learned label correlation structure, including in the missing-label setting (Zhu et al., 2017).

Label Correlation Grid (LCG) for label distribution learning computes a covariance matrix over labels, converts each covariance entry into a Gaussian-modeled vector-valued cell, and adds a label distribution projection algorithm based on KPCA + sub-MLP as a regularizer. The source is explicit that there is no defined quantity called projection correlation, projected correlation, or PCor; the correlation branch and projection branch are separate auxiliary targets rather than a unified projected-correlation operator (Guo et al., 2022).

LESC and gLESC treat label enhancement as correlation-preserving recovery: a self-expression coefficient matrix $Y$13 is learned from feature space, or jointly from feature and label space via tensor multi-rank minimization, and recovered label distributions are regularized by

$Y$14

or equivalently

$Y$15

The source states that these methods do not define PCor, but they provide a clear example of correlation-preserving label recovery under a learned sample-correlation operator (Zheng et al., 2020).

These comparisons clarify the scope of the term. In the precise sense established by the 2025 paper, Label Projection Correlation (PCor) is a dependence statistic between numerical features and categorical labels. Earlier projection- and correlation-based multi-label methods address different objects—sample geometry, label collaboration, latent labels, or label-distribution recovery—even when they are projection-based or correlation-aware. This suggests a useful distinction: PCor is a feature-label dependence coefficient, whereas the related multi-label methods are label-structure modeling frameworks.