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arxiv: 1906.09802 · v3 · pith:SZL7CWSSnew · submitted 2019-06-24 · 💻 cs.HC · cs.RO

Generalized Multiple Correlation Coefficient as a Similarity Measurements between Trajectories

Julen Urain , Jan Peters This is my paper

Pith reviewed 2026-05-25 17:28 UTC · model grok-4.3

classification 💻 cs.HC cs.RO
keywords generalized multiple correlation coefficienttrajectory similaritylinear transformation invarianceimitation learningcorrelation coefficienthuman-robot interactionmotion clustering
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The pith

The Generalized Multiple Correlation Coefficient extends the multiple correlation coefficient to measure trajectory similarity while remaining unchanged under any linear transformation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Generalized Multiple Correlation Coefficient as a similarity measure for trajectories that stays invariant under rotations, scalings, reflections, shears, and squeezes. It builds directly from the Pearson correlation coefficient and the coefficient of determination as a natural extension of the multiple correlation coefficient. The authors position this metric for use in human-robot interaction and imitation learning because prior measures lack full linear invariance, noise robustness, or real-time speed. If the extension works as described, trajectory clustering and comparison become possible without separate alignment steps for affine changes.

Core claim

The Generalized Multiple Correlation Coefficient is defined as the natural extension of the multiple correlation coefficient based on Pearson correlation and the coefficient of determination; this construction is claimed to produce a similarity value that equals one for any two trajectories related by an arbitrary linear transformation and remains robust under additive noise.

What carries the argument

The Generalized Multiple Correlation Coefficient (GMCC), constructed as the extension of the multiple correlation coefficient via Pearson's r and R-squared to enforce invariance under linear maps.

If this is right

  • Trajectories can be clustered directly in imitation learning without explicit preprocessing for scale, rotation or shear.
  • The metric supports real-time classification because it avoids iterative alignment procedures required by many existing distances.
  • Comparison remains stable when sensor noise is present because the underlying correlation formulation discounts additive perturbations.
  • The same coefficient applies to any pair of vector-valued time series without domain-specific adjustments for linear distortions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The invariance property may allow the metric to serve as a loss function inside gradient-based optimization for trajectory prediction models.
  • Because the construction uses only linear algebra operations already common in signal processing, integration into existing robot control pipelines requires minimal new code.
  • If the metric truly captures similarity independent of linear transforms, it could replace hand-crafted feature normalizations in datasets of human motion capture.

Load-bearing premise

Extending the multiple correlation coefficient in the stated way automatically produces invariance to every linear transformation while keeping the measure robust to noise.

What would settle it

Compute the GMCC between a trajectory and the same trajectory after applying a shear or squeeze matrix; the value must equal 1 if the invariance claim holds.

Figures

Figures reproduced from arXiv: 1906.09802 by Jan Peters, Julen Urain.

Figure 1
Figure 1. Figure 1: GMCC applied for clustering taught trajectories in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Artificially generated trajectories for testing the experiments in Section III-A [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distance Matrices computed for RV, dCor and GMCC [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dendrograms computed for RV, dCor and GMCC [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Similarity distance measure between two trajectories is an essential tool to understand patterns in motion, for example, in Human-Robot Interaction or Imitation Learning. The problem has been faced in many fields, from Signal Processing, Probabilistic Theory field, Topology field or Statistics field.Anyway, up to now, none of the trajectory similarity measurements metrics are invariant to all possible linear transformation of the trajectories (rotation, scaling, reflection, shear mapping or squeeze mapping). Also not all of them are robust in front of noisy signals or fast enough for real-time trajectory classification. To overcome this limitation this paper proposes a similarity distance metric that will remain invariant in front of any possible linear transformation.Based on Pearson Correlation Coefficient and the Coefficient of Determination, our similarity metric, the Generalized Multiple Correlation Coefficient (GMCC) is presented like the natural extension of the Multiple Correlation Coefficient. The motivation of this paper is two fold. First, to introduce a new correlation metric that presents the best properties to compute similarities between trajectories invariant to linear transformations and compare it with some state of the art similarity distances.Second, to present a natural way of integrating the similarity metric in an Imitation Learning scenario for clustering robot trajectories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes the Generalized Multiple Correlation Coefficient (GMCC) as a trajectory similarity metric extending the multiple correlation coefficient via Pearson correlation and the coefficient of determination. It claims GMCC is invariant under all linear transformations (rotation, scaling, reflection, shear, squeeze) and robust to noise, while being suitable for real-time use; the work also compares it to existing metrics and integrates it into an imitation-learning pipeline for clustering robot trajectories.

Significance. If the invariance property is rigorously established, GMCC would address a genuine gap in trajectory similarity measures for HRI and imitation learning by handling a broader class of transformations than current methods while remaining computationally lightweight. The explicit integration into a clustering step for robot trajectories supplies a concrete application context.

major comments (2)
  1. [Abstract] Abstract: The claim that GMCC(X,Y) remains invariant under arbitrary invertible linear maps A (explicitly including shear [[1,a],[0,1]] and squeeze) is asserted without derivation. While multiple correlation is invariant to reparameterization of the regressors, the manuscript supplies no argument showing that the joint transformation of both trajectories by the same A preserves the coefficient when the map is non-orthogonal.
  2. [Abstract] Abstract: No explicit definition, equation, or proof is given for GMCC itself, nor any verification (analytic or numeric) that the construction is independent of the coordinate basis after a general linear change. This absence makes the central invariance claim impossible to evaluate from the supplied text.
minor comments (2)
  1. [Title] Title: 'Measurements' should be 'Measurement'.
  2. [Abstract] Abstract: Repeated use of 'in front of' for 'under' or 'to' is nonstandard; 'similarity distance metric' is internally inconsistent since a correlation coefficient is not a distance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that GMCC(X,Y) remains invariant under arbitrary invertible linear maps A (explicitly including shear [[1,a],[0,1]] and squeeze) is asserted without derivation. While multiple correlation is invariant to reparameterization of the regressors, the manuscript supplies no argument showing that the joint transformation of both trajectories by the same A preserves the coefficient when the map is non-orthogonal.

    Authors: We agree that the abstract asserts the invariance property without supplying a derivation. In the revised manuscript we will add an explicit proof that GMCC(X,Y) is unchanged when both trajectories are transformed by the same invertible linear map A, including non-orthogonal cases such as shear and squeeze. The argument will start from the definition of the multiple correlation coefficient and show that the joint linear transformation leaves the coefficient invariant. revision: yes

  2. Referee: [Abstract] Abstract: No explicit definition, equation, or proof is given for GMCC itself, nor any verification (analytic or numeric) that the construction is independent of the coordinate basis after a general linear change. This absence makes the central invariance claim impossible to evaluate from the supplied text.

    Authors: We acknowledge that the abstract contains neither the explicit definition of GMCC nor any derivation or verification of its invariance. The revised manuscript will include the mathematical definition of GMCC (as the natural extension of the multiple correlation coefficient via Pearson correlation and the coefficient of determination) together with both an analytic proof and numerical verification that the measure is independent of the coordinate basis after an arbitrary invertible linear change. revision: yes

Circularity Check

0 steps flagged

GMCC defined directly from Pearson correlation and R² without self-referential reduction

full rationale

The paper introduces GMCC explicitly as the natural extension of the Multiple Correlation Coefficient constructed from the Pearson Correlation Coefficient and the Coefficient of Determination. No load-bearing step reduces the claimed linear-invariance property to a fitted parameter, a self-citation chain, or an input by construction; the definition is presented as a direct statistical generalization whose properties are asserted to follow from the base coefficients. The derivation therefore remains self-contained against external statistical benchmarks rather than internally circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore limited to statements appearing in the abstract.

axioms (1)
  • domain assumption GMCC is the natural extension of the Multiple Correlation Coefficient that remains invariant under any linear transformation
    Stated directly in the abstract as the core proposal

pith-pipeline@v0.9.0 · 5732 in / 1149 out tokens · 40540 ms · 2026-05-25T17:28:44.641690+00:00 · methodology

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