Dyadic Distance Matrix: Analysis & Applications

Updated 5 July 2026

Dyadic Distance Matrix (DDM) is a matrix representation organizing pairwise distances between dyadic elements, crucial for analyzing conversational interactions and metric spaces.
It captures turn-level similarities using cosine distances, preserving global dialogue patterns and exposing coadaptation between speakers.
Beyond conversation, DDM is applied in diffusion geometry, graph summaries, and latent factor models, offering versatile insights into complex dyadic relations.

Dyadic Distance Matrix (DDM) most directly denotes a matrix representation of pairwise distances defined on dyads. In current arXiv usage, the term is explicit in work on dyadic conversational interaction, where a two-speaker dialogue is mapped to a two-dimensional matrix whose $(i,j)$ entry is the cosine distance between the $i$ -th turn of one speaker and the $j$ -th turn of the other (Nilabh et al., 1 Jun 2026). Related literatures use closely adjacent constructions rather than a single uniform term: finite matrices of the dyadic metric $\delta(x,y)$ on $\mathbb{R}^+$ (Acosta et al., 2023), learned distance or similarity structures induced by models for dyadic data (Chen et al., 2016), and graph-theoretic summaries of pairwise shortest-path distances (Roginski et al., 2015). The term is therefore context-dependent; what unifies these uses is the organization of pairwise relations between the two elements of a dyad into a matrix whose geometry is itself analytically informative.

1. Conversational DDM as an explicit matrix representation

In dyadic dialogue modeling, the DDM is the central representation for encoding conversational interaction over an entire two-speaker exchange. Let $\mathbf{a}_i \in \mathbb{R}^d$ denote the embedding of the $i$ -th turn of speaker $A$ , and let $\mathbf{b}_j \in \mathbb{R}^d$ denote the embedding of the $j$ -th turn of speaker $i$ 0. The matrix is defined by

$i$ 1

Rows are the turns of speaker $i$ 2 in temporal order, columns are the turns of speaker $i$ 3 in temporal order, and each cell is the cosine distance between one turn of $i$ 4 and one turn of $i$ 5 (Nilabh et al., 1 Jun 2026).

This construction is explicitly a distance matrix rather than a similarity matrix. Lower values indicate greater cross-speaker similarity, and higher values indicate greater dissimilarity. Because the matrix records all cross-speaker turn pairs rather than only adjacent turns, it encodes long-range dependencies across the full dialogue. The representation is therefore global in scope and preserves an interactional geometry that can be treated as a two-dimensional signal or image (Nilabh et al., 1 Jun 2026).

The intended theoretical role of the conversational DDM is to separate dyad-specific interactional structure from speaker-specific traits. Prior entrainment measures often rely on adjacent-turn comparisons or speaker-level aggregates. By contrast, the DDM retains the full cross-product of turns and thereby exposes patterns that require both speakers jointly, rather than properties attributable to either speaker in isolation. A plausible implication is that the DDM functions as a representation of coadaptation rather than merely a summary of speaker identity.

2. Construction pipeline and representational layers

The DDM construction pipeline begins with turn segmentation. On the CANDOR corpus, conversations are processed with Cliffhanger sentence-level segmentation to obtain semantically coherent speaker turns, and conversations with fewer than 20 turns per speaker are discarded (Nilabh et al., 1 Jun 2026). This yields two temporally ordered turn sequences,

$i$ 6

Each turn is then embedded in one of four representational spaces. The paper uses wav2vec 2.0, with mean-pooled final transformer-layer outputs, to produce $i$ 7-dimensional turn vectors; openSMILE with the GeMAPS configuration to produce $i$ 8-dimensional acoustic descriptors; x-vector embeddings from a pre-trained ECAPA-TDNN model via pyannote.audio to produce $i$ 9-dimensional speaker representations; and all-MiniLM to produce $j$ 0-dimensional sentence embeddings (Nilabh et al., 1 Jun 2026). These correspond to acoustic, structural, and semantic representational layers.

To reduce static speaker-specific biases before matrix construction, per-speaker $j$ 1-normalization is applied to wav2vec 2.0, x-vector, and openSMILE embeddings. For a speaker $j$ 2 with mean $j$ 3 and standard deviation $j$ 4, each embedding $j$ 5 is normalized as

$j$ 6

The stated purpose is to reduce global speaker tendencies so that the DDM emphasizes interactional variation rather than fixed vocal baselines (Nilabh et al., 1 Jun 2026).

After pairwise cosine distances are computed, each DDM is resized to a fixed $j$ 7 resolution for classification. The matrices are then standardized to unit variance (Nilabh et al., 1 Jun 2026). This normalizes variable-length conversations into a common tensor shape while retaining the original definition of the DDM on the true number of turns. In practical terms, the resulting object is a real-valued image-like matrix representing cross-speaker distances over the entire conversation.

3. Speaker-switch evaluation and empirical behavior

The principal validation protocol for the conversational DDM is the speaker-switch test. Given an original conversation $j$ 8 and an unrelated conversation $j$ 9, the original DDM is constructed from $\delta(x,y)$ 0 and $\delta(x,y)$ 1, whereas the switched DDM retains $\delta(x,y)$ 2 and replaces $\delta(x,y)$ 3 with $\delta(x,y)$ 4:

$\delta(x,y)$ 5

This preserves turn-level statistics while disrupting the original dyadic coadaptation (Nilabh et al., 1 Jun 2026).

The binary task is to classify DDMs as real or switched. The evaluated classifiers are ResNet-50, a shallow three-layer CNN, and an MLP on flattened $\delta(x,y)$ 6 inputs. Training uses Adam with learning rate $\delta(x,y)$ 7, batch size $\delta(x,y)$ 8, early stopping on validation loss, and a $\delta(x,y)$ 9 train/validation/test split stratified by conversation. Performance is reported with accuracy, Macro-F1, and equal error rate (Nilabh et al., 1 Jun 2026).

On CANDOR, semantic DDMs built from all-MiniLM are almost perfectly discriminable. ResNet-50 achieves Acc $\mathbb{R}^+$ 0, F1 $\mathbb{R}^+$ 1, and EER $\mathbb{R}^+$ 2; CNN and MLP also perform strongly, with Acc $\mathbb{R}^+$ 3 and $\mathbb{R}^+$ 4 respectively (Nilabh et al., 1 Jun 2026). Acoustic and structural DDMs are more difficult. With ResNet-50, wav2vec 2.0 reaches Acc $\mathbb{R}^+$ 5, x-vector reaches Acc $\mathbb{R}^+$ 6, and openSMILE reaches Acc $\mathbb{R}^+$ 7, whereas simpler models are weaker and some MLP settings fall to chance-like behavior (Nilabh et al., 1 Jun 2026). These results show that the DDM contains interaction-relevant structure even when that structure is not linearly separable.

Cross-corpus comparisons with LibriSpeech yield markedly higher discriminability in read speech. For LibriSpeech Switch, x-vector DDMs reach Acc $\mathbb{R}^+$ 8 and EER $\mathbb{R}^+$ 9 for all tested classifiers, while wav2vec 2.0, openSMILE, and all-MiniLM also attain very high performance (Nilabh et al., 1 Jun 2026). The reported interpretation is that read speech exhibits lower prosodic variability, making mismatched pairings easier to detect; naturalistic conversation in CANDOR is harder because prosodic and stylistic adaptation is more complex.

GradCAM analyses reveal modality-specific structural signatures. For semantic DDMs, real conversations show strong activation along and near the matrix diagonal, consistent with temporally proximate topical alignment; switched conversations lack this diagonal structure (Nilabh et al., 1 Jun 2026). Acoustic and structural DDMs produce more distributed activations, which the paper associates with longer-range prosodic accommodation rather than strictly local adjacency. This supports the claim that the DDM is not merely a container for scalar distances but a structured object whose spatial organization carries interactional information.

4. Dyadic metric matrices and diffusion geometry

A distinct mathematical use of dyadic distance arises on $\mathbf{a}_i \in \mathbb{R}^d$ 0. There, the dyadic distance is defined by

$\mathbf{a}_i \in \mathbb{R}^d$ 1

For a finite set of points $\mathbf{a}_i \in \mathbb{R}^d$ 2, one obtains the dyadic distance matrix

$\mathbf{a}_i \in \mathbb{R}^d$ 3

where $\mathbf{a}_i \in \mathbb{R}^d$ 4 is the smallest dyadic interval containing both points (Acosta et al., 2023).

This matrix is tied to the fractional dyadic Laplacian

$\mathbf{a}_i \in \mathbb{R}^d$ 5

and to the associated diffusion kernel

$\mathbf{a}_i \in \mathbb{R}^d$ 6

where $\mathbf{a}_i \in \mathbb{R}^d$ 7 is the Haar wavelet system (Acosta et al., 2023). The corresponding Coifman-Lafon diffusion distance is

$\mathbf{a}_i \in \mathbb{R}^d$ 8

The central structural theorem is that, for each fixed $\mathbf{a}_i \in \mathbb{R}^d$ 9, the diffusion distance is a function of the dyadic distance:

$i$ 0

Consequently, if $i$ 1 denotes the finite diffusion distance matrix on points $i$ 2, then

$i$ 3

so the diffusion geometry is obtained by entrywise application of a scalar function to the dyadic distance matrix (Acosta et al., 2023). Even though these metrics are not equivalent, the families of balls are the same: the balls in the diffusion metric coincide with dyadic intervals. In this setting, a DDM is not a conversational object but the finite matrix realization of an ultrametric-like dyadic geometry.

5. Induced and summary forms in dyadic data and graphs

In dyadic data prediction, the phrase “Dyadic Distance Matrix” is not used explicitly, but the Heterogeneous Matrix Factorization model furnishes a direct route to such constructions. The observed dyadic matrix $i$ 4 is modeled as

$i$ 5

with latent user factors $i$ 6 and item factors $i$ 7, DP-based user and item communities, and variational Bayesian as well as online variational inference (Chen et al., 2016). From posterior means, one can define user-item similarities $i$ 8, user-user distances $i$ 9, item-item distances, and dyad-dyad distances on concatenated latent vectors (Chen et al., 2016). This suggests a probabilistic DDM interpretation in which the matrix is learned from sparse dyadic observations and inherits bicluster structure from the model’s nonparametric communities.

Graph theory offers a different summary construction. For a simple, connected, undirected, unweighted graph with diameter $A$ 0, the neighbor matrix is

$A$ 1

Its first column is the degree sequence, and each subsequent column counts distance- $A$ 2 neighbors (Roginski et al., 2015). The matrix is explicitly presented as a summary of the graph distance matrix: it records, for each vertex, the histogram of pairwise geodesic distances to all other vertices. The paper therefore describes it as a summary of vertex-vertex distance distributions and shows that it contains numerous graph statistics, including radius, center, closeness centrality, average distance, periphery, and information about graph powers (Roginski et al., 2015).

In this graph setting, calling the neighbor matrix a DDM would be an interpretation rather than the paper’s formal terminology. That interpretation is well motivated, because row $A$ 3 records how many dyads $A$ 4 fall into each distance shell. The matrix preserves the distribution of dyadic distances while discarding the identity of which particular vertex occupies each shell. A plausible implication is that the neighbor matrix is a compressed distance-profile DDM: richer than the degree sequence, but less informative than the full distance matrix.

Several adjacent literatures use “dyadic” and “DDM” in ways that are technically important but not synonymous with Dyadic Distance Matrix.

Context	Matrix object	Relation to “DDM”
Dyadic conversational interaction	Cross-speaker turn-turn cosine distance matrix $A$ 5	Explicit use of Dyadic Distance Matrix
Fractional dyadic Laplacian	Finite matrix $A$ 6	Finite dyadic distance matrix
Dyadic data prediction	User-user, item-item, or dyad-dyad distances from latent factors	Phrase not explicit; induced construction
Graph topology	Neighbor matrix $A$ 7	Distance-profile summary over dyads
Sparse SPD factorization	Permuted dyadic form $A$ 8 and distance-like packing matrices	Reconstructed DDM sense
Radar systems	DDM phase codes	DDM means Doppler-Division Multiplexing

In sparse positive definite matrix analysis, dyadic structure refers to recursive dyadic sparsity patterns rather than pairwise conversational or metric distances. The framework defines horizontal, vertical, and symmetric dyadic matrix classes such as $A$ 9, shows that a symmetric positive definite matrix can be packed into dyadic form, and derives efficient inversion via a sparse dyadic Gram-Schmidt orthogonalization. The paper explicitly reconstructs a DDM-like notion as a symmetric positive-definite matrix that can be permuted into a symmetric dyadic structure and is associated with distance-like matrices used in the packing step (Kos et al., 13 May 2025). This is a mathematically substantive, but reconstructed, usage.

In cryptography, “dyadic matrices” denote structured MDS matrices related to dyadic codes and used in block-cipher diffusion layers. The abstract of “Construction of dyadic MDS matrices for cryptographic applications” describes classical MDS matrices that are “strong symmetric,” related to dyadic codes, and amenable to efficient evaluation or circuit implementation (Berger, 2014). These objects optimize diffusion and implementation properties; they are not distance matrices.

In automotive radar, “DDM” has an entirely different meaning: Doppler-Division Multiplexing. “Relaxed Multi-Tx DDM Online Calibration” studies DDM phase-code sequences for FMCW MIMO radar and explicitly states that the paper does not use “Dyadic Distance Matrix” at all (Jeannin et al., 2024). This disambiguation is necessary because the acronym is shared across unrelated technical domains.

The principal misconception surrounding DDM is therefore terminological rather than mathematical. There is no single, field-independent object called a Dyadic Distance Matrix. The phrase is explicit and operationally central in dyadic conversational interaction (Nilabh et al., 1 Jun 2026), mathematically natural for finite matrices of the dyadic metric on $\mathbf{b}_j \in \mathbb{R}^d$ 0 (Acosta et al., 2023), and suggestive in dyadic data analysis, graph topology, and sparse matrix factorization (Chen et al., 2016, Roginski et al., 2015, Kos et al., 13 May 2025). Across these settings, the common theme is the matrix organization of pairwise dyadic relations; the underlying entities, metrics, and analytic purposes differ substantially.