Orthonormal Density-Ratio Decomposition

• Orthonormal density‐ratio decomposition is a Hilbert-space representation that expresses relative data via mutually orthogonal, normalized log‐ratio coordinates.
• It spans diverse settings—such as the Aitchison simplex, Bayes spaces, and spectral methods—to tailor the decomposition for compositional and dependence modeling.
• Applications range from microbiome and single-cell analysis to density estimation, providing stable, interpretable features through additive variance decomposition.

Searching arXiv for the cited papers and adjacent literature on orthonormal density-ratio decomposition, Bayes spaces, and tree-structured log-ratio methods. Orthonormal density-ratio decomposition denotes a class of representations in which a composition, probability vector, or probability density is expressed through mutually orthogonal, normalized coordinates built from density ratios or log-density ratios in a geometry where only relative information is meaningful. In contemporary usage, the term covers several technically distinct but structurally related settings: isometric log-ratio coordinates on the Aitchison simplex, clr-based decompositions in Bayes spaces, spectral decompositions of the dependence ratio $\rho(x,y)=p_{XY}(x,y)/(p_X(x)p_Y(y))$, and orthonormal polynomial expansions of a target density relative to a reference density (Yamada et al., 10 Jun 2026, Czolková et al., 17 Jun 2026, Ma et al., 2024, Asmussen et al., 2016).

1. Conceptual scope

In all of these settings, the primary object is not an absolute density level but a relative quantity: a componentwise ratio, a log-contrast, or a likelihood ratio with respect to a reference measure. Orthogonality then refers to a Hilbert-space inner product induced by the relevant geometry. Normalization ensures unit norm, so the resulting coordinates or basis functions are directly comparable and can be aggregated without double counting.

The principal variants differ by the underlying sample space and by the ratio being decomposed. On the simplex, the object is a composition $x\in\Delta^{d-1}$ and the coordinates are orthonormal log-ratios under the Aitchison inner product. In Bayes spaces, the object is an equivalence class of positive densities modulo proportionality, and the clr transform converts multiplicative density geometry into linear $L^2$ geometry. In dependence modeling, the object is the ratio of a joint density to the product of marginals, and the decomposition takes the form of a singular-system expansion. In orthogonal-polynomial methods, the object is $f/f_\nu$, expanded in a basis orthonormal with respect to the reference density $f_\nu$ (Yamada et al., 10 Jun 2026, Czolková et al., 17 Jun 2026, Ma et al., 2024, Szabłowski, 2010).

Setting	Ratio object	Canonical coordinates
Aitchison simplex	log-ratios of parts	ILR or tree-aligned ILR
Bayes space	centered log-density ratios	clr coordinates in $L_0^2$
Dependence operator	$p_{XY}/(p_Xp_Y)$	eigenfunctions and eigenvalues
Polynomial expansion	$f/f_\nu$	orthonormal polynomial coefficients

A recurrent misconception is to treat these constructions as interchangeable. They are not. Each depends on a specific ambient Hilbert structure: the Aitchison inner product for compositional vectors, the Bayes inner product for densities modulo scale, the $L^2(p_X)\otimes L^2(p_Y)$ structure for dependence ratios, or the $L^2(\nu)$ geometry defined by a chosen reference density. What is shared is the relative, scale-invariant viewpoint.

2. Geometric foundations

For compositional data, the natural domain is the open simplex

$x\in\Delta^{d-1}$0

where only ratios $x\in\Delta^{d-1}$1 are informative. Aitchison geometry equips this simplex with perturbation and powering, and with the inner product

$x\in\Delta^{d-1}$2

The clr transform maps the simplex isometrically to the zero-sum hyperplane

$x\in\Delta^{d-1}$3

via

$x\in\Delta^{d-1}$4

Any orthonormal basis $x\in\Delta^{d-1}$5 of $x\in\Delta^{d-1}$6 yields an ILR transform

$x\in\Delta^{d-1}$7

which is an isometry from the simplex to $x\in\Delta^{d-1}$8 (Yamada et al., 10 Jun 2026).

For general densities, Bayes spaces formalize the same “only ratios matter” principle. Two positive densities are equivalent when they are proportional, and the square-log-integrable Bayes space $x\in\Delta^{d-1}$9 is a Hilbert space under perturbation and powering. Its clr transform is

$L^2$0

with image in

$L^2$1

The clr map is a linear isometric isomorphism, so orthogonality in Bayes space is equivalent to orthogonality in $L^2$2 of centered log-density functions (Czolková et al., 17 Jun 2026). In the bivariate case, the same structure appears in $L^2$3, where the inner product depends only on differences of log-densities and the clr transform produces zero-integral log-density representations in $L^2$4 (Hron et al., 2020).

These geometric facts determine what “orthonormal density-ratio decomposition” means in practice: a coordinate system or basis must respect the zero-sum or zero-integral constraint, must be orthonormal in the induced Hilbert geometry, and must reconstruct the original composition or density up to the appropriate closure or normalization operation.

3. Tree-structured log-ratio decompositions on the simplex

A recent discrete realization is PolyILR, a canonical orthonormal decomposition of the Aitchison tangent space aligned with an arbitrary rooted tree on the components. The leaves are the components of the composition; each internal node $L^2$5 has $L^2$6 children, with descendant clades $L^2$7 and sizes $L^2$8. PolyILR defines the local contrast subspace

$L^2$9

together with the weighted inner product

$f/f_\nu$0

At each node, a canonical Helmert matrix is orthonormalized by weighted Gram–Schmidt, and the resulting local contrasts are spread to leaf-level vectors by assigning a constant coefficient within each child clade. The resulting global matrix $f/f_\nu$1 satisfies

$f/f_\nu$2

so $f/f_\nu$3 is an ILR transform. Every coordinate is indexed by a node-location pair $f/f_\nu$4, and each coordinate is a log-contrast of geometric means over the child clades of a specific node (Yamada et al., 10 Jun 2026).

This construction yields a complete and multiscale decomposition. The dimension count is

$f/f_\nu$5

so no degrees of freedom are lost at polytomous nodes. Contrasts attached to different internal nodes are orthogonal, which induces an orthogonal direct-sum decomposition of ILR space by tree location. In the binary case, PolyILR reduces exactly to PhILR’s balance formula; for non-binary trees it avoids arbitrary binarization, which the paper identifies as a source of severe instability in feature selection for PhILR (Yamada et al., 10 Jun 2026).

The density-ratio interpretation is explicit. If $f/f_\nu$6, then for a coordinate attached to node $f/f_\nu$7,

$f/f_\nu$8

where $f/f_\nu$9 is the geometric mean over the $f_\nu$0-th child clade. Each coordinate is therefore a normalized log-ratio of geometric mean densities between clades. The same paper establishes a connection with softmax classifiers: if $f_\nu$1, then

$f_\nu$2

so logits modulo additive shifts and CLR coordinates of probabilities inhabit the same quotient geometry. A tree-aligned basis can therefore be applied directly to logits, yielding a decomposition of prediction errors and gradients by hierarchical contrast directions (Yamada et al., 10 Jun 2026).

Empirically, PolyILR was illustrated on microbiome and single-cell compositions. The reported examples include “Streptococcaceae vs Lactobacillus + Leuconostocaceae,” “Lactococcus vs Streptococcus,” “Prevotella vs [Bacteroides + Alistipes + ...],” and “Myeloid cell vs Erythrocyte/Megakaryocyte + Hematopoietic precursor cell.” The paper states that PolyILR features are as predictive as CLR/PhILR, with “same SVM/LR accuracy; RF differences are modest,” and that feature selection based on PolyILR coordinates is highly stable across cross-validation. Construction is done once per tree, with worst-case complexity $f_\nu$3 for a star tree and often closer to $f_\nu$4 for balanced trees (Yamada et al., 10 Jun 2026).

4. Bayes-space decompositions of densities

In Bayes spaces, orthonormal density-ratio decomposition takes the form of orthogonal projection of clr-transformed densities onto geometrically meaningful subspaces. For multivariate densities on a product space with reference measure $f_\nu$5, the central objects are the geometric marginals and the interaction spaces. For any non-empty $f_\nu$6, the subspace $f_\nu$7 contains densities whose information is confined to coordinates in $f_\nu$8, and the geometric marginal on $f_\nu$9 is the unique orthogonal projection onto that subspace. The multivariate paper then defines the independence space as the direct sum of univariate marginal spaces and defines higher-order interaction spaces as orthogonal complements of lower-order effects. The resulting theorem states that any $L_0^2$0-variate density $L_0^2$1 has a unique orthogonal decomposition

$L_0^2$2

yielding $L_0^2$3 orthogonal components: $L_0^2$4 geometric marginals and $L_0^2$5 pure interaction components (Czolková et al., 17 Jun 2026).

The clr representation makes the decomposition especially transparent: $L_0^2$6 Because the subspaces are orthogonal, the sample Bayes covariance matrix of the decomposed components is diagonal, the component variances add to total variance, and the decomposition is “PCA-optimal” in the sense stated in the paper: principal components are simply the individual orthogonal components ordered by their variance (Czolková et al., 17 Jun 2026). The same work further shows that FPCA on raw multivariate densities and multivariate FPCA on the vector of orthogonal components are equivalent: eigenfunctions for the full density decompose as orthogonal sums of marginal and interaction contributions, and scores decompose additively across those contributions.

The bivariate Bayes-space construction gives the simplest concrete version. For $L_0^2$7, the clr geometric marginals are obtained by integrating $L_0^2$8 over one coordinate, yielding $L_0^2$9 and $p_{XY}/(p_Xp_Y)$0. The independent part is

$p_{XY}/(p_Xp_Y)$1

and the interactive part is

$p_{XY}/(p_Xp_Y)$2

equivalently $p_{XY}/(p_Xp_Y)$3 in Bayes-space sense. The decomposition is orthogonal: $p_{XY}/(p_Xp_Y)$4 and in fact $p_{XY}/(p_Xp_Y)$5, $p_{XY}/(p_Xp_Y)$6, and $p_{XY}/(p_Xp_Y)$7 are mutually orthogonal. This yields Pythagorean identities for the norms and motivates the simplicial deviance

$p_{XY}/(p_Xp_Y)$8

which measures the fraction of total clr “energy” attributable to dependence (Hron et al., 2020).

That paper also provides a tensor-product B-spline implementation. A clr surface is represented by a spline with coefficients constrained to satisfy the zero-integral condition required by clr geometry. The independent spline component is the sum of the univariate clr-marginal splines, and the interactive spline component is the residual coefficient tensor. This furnishes a finite-dimensional computational realization of the orthogonal decomposition of log-density ratios (Hron et al., 2020).

5. Spectral and polynomial decompositions

A different usage arises in dependence estimation between paired random variables. If $p_{XY}/(p_Xp_Y)$9 and $f/f_\nu$0 have joint density $f/f_\nu$1 and marginals $f/f_\nu$2, the ratio

$f/f_\nu$3

equals $f/f_\nu$4 under independence and can be treated as a positive definite kernel on $f/f_\nu$5. Under the assumptions used in the cortico-muscular paper, this yields a spectral decomposition

$f/f_\nu$6

where $f/f_\nu$7 and $f/f_\nu$8 are orthonormal systems in $f/f_\nu$9 and $L^2(p_X)\otimes L^2(p_Y)$0, and $L^2(p_X)\otimes L^2(p_Y)$1 quantify dependence strength. The paper interprets this as the infinite-dimensional analogue of an SVD, and proposes FMCA-LD and FMCA-T for learning the dominant eigensystem from samples through neural feature maps $L^2(p_X)\otimes L^2(p_Y)$2 and $L^2(p_X)\otimes L^2(p_Y)$3. FMCA-LD minimizes a log-determinant objective corresponding to $L^2(p_X)\otimes L^2(p_Y)$4; FMCA-T minimizes

$L^2(p_X)\otimes L^2(p_Y)$5

which directly maximizes $L^2(p_X)\otimes L^2(p_Y)$6. Whitening and SVD of the empirical cross-correlation recover orthonormal approximations to the eigenfunctions and a finite-rank approximation to $L^2(p_X)\otimes L^2(p_Y)$7 (Ma et al., 2024).

A closely related but older family of methods expands a density ratio relative to a fixed reference density $L^2(p_X)\otimes L^2(p_Y)$8. If $L^2(p_X)\otimes L^2(p_Y)$9 and $L^2(\nu)$0 is an orthonormal polynomial basis in $L^2(\nu)$1, then

$L^2(\nu)$2

and hence

$L^2(\nu)$3

This is an orthogonal projection of the density ratio in $L^2(\nu)$4. The framework was developed for normal, gamma, and lognormal references, with Hermite, Laguerre, and explicit lognormal orthonormal polynomials, respectively. The normal and gamma systems are complete under stated exponential-integrability conditions, but the lognormal case is not: the paper proves that the orthonormal polynomials are not dense in $L^2(\nu)$5, relating the failure to moment indeterminacy of the lognormal law (Asmussen et al., 2016).

The orthogonal-polynomial viewpoint also appears in explicit density expansions between special-function families. One density can be written as another density multiplied by an orthogonal polynomial series, so the density ratio itself becomes the object of orthogonal expansion. The paper on $L^2(\nu)$6-families develops this for the $L^2(\nu)$7-Normal, Rogers, Al-Salam–Chihara, Chebyshev, and related densities. A central example is the Poisson–Mehler kernel,

$L^2(\nu)$8

which becomes an orthonormal expansion after normalization of the $L^2(\nu)$9-Hermite basis. The reciprocal kernel is likewise expanded in the orthogonal basis attached to the conditional density. Proposition 1 of that paper gives a general rule: if $x\in\Delta^{d-1}$00 has an expansion in the polynomial system orthogonal with respect to $x\in\Delta^{d-1}$01, then one can construct related polynomial systems and finite-step connection relations between the two measures (Szabłowski, 2010).

These spectral and polynomial constructions differ from ILR and Bayes-space methods in that the object being decomposed is usually a ratio between two distributions or two measures rather than a single composition. The common feature remains orthonormalization with respect to a geometry induced by the reference law.

6. Interpretation, applications, and limitations

The practical value of orthonormal density-ratio decomposition is interpretability under an additive variance or energy accounting. On the simplex, node-indexed coordinates permit aggregation by depth, subtree, or branch because orthogonal node subspaces can be summed without double counting. In Bayes spaces, geometric marginals and interaction terms separate independent structure from dependence structure, and total variance decomposes by Pythagoras. In dependence-ratio methods, eigenvalues order the modes of shared structure, while the associated eigenfunctions provide feature spaces aligned with those modes (Yamada et al., 10 Jun 2026, Czolková et al., 17 Jun 2026, Ma et al., 2024).

Applications in the cited literature are correspondingly diverse. PolyILR was evaluated on microbiome and single-cell benchmarks, where it produced stable interpretable features and multiscale inference aligned with taxonomy or ontology; the paper specifically highlights HMP body sites, westernization in cMD3, and single-cell leukemia versus healthy controls (Yamada et al., 10 Jun 2026). Bayes-space decompositions were illustrated on housing and geological empirical data in the multivariate setting and on anthropometric age-group densities in the bivariate setting, where the interaction component tracked life-course changes in dependence and could be compared with Spearman and Kendall summaries (Czolková et al., 17 Jun 2026, Hron et al., 2020). The dependence-ratio operator framework was applied to EEG/EMG data, where learned eigenfunctions classified movements and subjects and revealed channel and temporal dependencies confirming activation of specific EEG channels during movement (Ma et al., 2024). Orthonormal polynomial expansions were used for approximating densities of sums of lognormals, including transformed or exponentially tilted variants (Asmussen et al., 2016).

Two limitations recur. First, orthogonality is geometry-dependent. CLR coordinates, ILR coordinates, Bayes-space decompositions, and dependence-ratio eigensystems are not interchangeable summaries, because each is orthonormal with respect to a different inner product. Second, completeness is not automatic. PolyILR produces a full basis on the simplex because $x\in\Delta^{d-1}$02, and the Bayes-space constructions produce full orthogonal decompositions by theorem, but polynomial density-ratio expansions can fail to span the ambient $x\in\Delta^{d-1}$03 space, as in the lognormal reference case (Yamada et al., 10 Jun 2026, Czolková et al., 17 Jun 2026, Asmussen et al., 2016).

A further issue concerns structural choices. PolyILR is canonical only up to child ordering and sign convention, and the paper emphasizes that arbitrary binary resolution of polytomies changes coordinates and their interpretations. In the dependence-ratio setting, neural estimation introduces approximation and optimization error even though the matrix objectives are tied to the target spectrum. In spline-based Bayes methods, computational practice depends on basis choice, smoothing penalties, and enforcement of the clr zero-integral constraint (Yamada et al., 10 Jun 2026, Ma et al., 2024, Hron et al., 2020).

Taken together, the literature identifies orthonormal density-ratio decomposition as a family of Hilbert-space representations for relative data. Whether the domain is the simplex, a space of densities modulo scale, a paired-view dependence operator, or a density ratio relative to a reference law, the central aim is the same: to replace raw densities by orthonormal coordinates that preserve the ratio geometry and expose interpretable structure at the correct resolution.