Introduction

Representation learning studies how high-dimensional observations can be mapped to lower-dimensional coordinates that preserve the information needed for a task. This premise appears throughout modern machine learning: images, graphs, sequences, and physical measurements often contain repeated structure, nuisance variation, and symmetries that should not change the relevant prediction. A useful representation should therefore discard irrelevant variation while retaining the information required for inference.

Classical statistics gives a precise language for this goal. A statistic is sufficient for a parameter when it preserves all information in the data relevant to estimating that parameter. It is minimally sufficient when no strictly coarser statistic preserves the same information. In representation learning, the analogous question is geometric: when can a learned representation be said to contain exactly the task-relevant structure of the data, and no more?

This paper studies that question for data with group-structured symmetries. Suppose a group $G$ acts on a sample space $X$. Points in the same orbit represent different observations that remain unchanged under a particular group action. If the task is invariant under this action, then these observations should not be distinguished by a minimal representation. The natural candidate is the quotient map $\pi:X\to X/G,$ which collapses each group orbit to a single equivalence class. We show that, under an invariant likelihood or conditional target law, this quotient is not just an invariant representation: it is a sufficient representation. Under an additional orbit-separation condition, it is minimally sufficient.

This connects statistical sufficiency to geometric factorization. The quotient $X/G$ removes precisely the variation generated by the group action, while retaining the variation specifically relevant to the task at-hand. In this sense, minimal representation learning can be viewed as the problem of identifying and collapsing nuisance symmetries without destroying information needed for prediction.

We analyze compact Lie groups and homogeneous spaces, where harmonic analysis provides canonical coordinates for group-structured data. By the Peter-Weyl theorem, square-integrable functions on a compact group decompose into matrix coefficients of irreducible unitary representations. For finite-band harmonic exponential families, these generalized Fourier coefficients become the natural sufficient statistics. We prove that, under the standard full-rank conditions for exponential families, the empirical generalized Fourier coefficients are minimally sufficient for the model parameters.

We then study the normalization constant of these harmonic exponential families. Although finite-band log-densities are tractable to write down, their exponentials generally generate higher-order harmonics, making the partition function difficult to compute. We show that the Clebsch-Gordan decomposition expresses the normalization constant algebraically: after expanding the exponential, tensor products of irreducible representations decompose into direct sums, and Haar integration retains only the trivial representation component. Thus the partition function can be represented as the trivial-irrep projection of the exponential tensor algebra.

Finally, we define the conditions under which a representation is minimal sufficient for an entire class of tasks. This becomes important for multi-task and contextual learning, where the representation need generalize to a larger range of problems. While current works in representation learning focus on performance metrics (ie. loss) and information-theoretic lower bounds, we define completeness geometrically as the intersection of minimal sufficient representations over the range of tasks. This provides a structural and geometric analysis of minimal sufficiency that supports AI safety and interpretability.

Contributions

The paper therefore makes four contributions.

1. First, it formulates minimal sufficient representation learning as a quotient problem under group-invariant statistical structure.
2. Second, it connects this quotient view to harmonic sufficient statistics on compact groups and homogeneous spaces.
3. Third, it gives an algebraic representation of the harmonic exponential-family partition function using Clebsch-Gordan decomposition.
4. Fourth, we define task-completeness, where the representation is invariant to the action of an entire family of tasks.

Related Work

Exponential Families and Sufficient Statistics

Exponential families provide the classical statistical setting in which sufficiency is most explicit. Given a dominated family of distributions $p_\theta(x)$, a statistic $T(X)$ is sufficient for $\theta$ if the likelihood depends on the sample only through $T(X)$ [casella_statistical_2024] [hipp_sufficient_1974]. In a full-rank exponential family, the empirical sum of the natural statistics is not only sufficient but minimally sufficient under standard regularity conditions. This makes exponential families a natural starting point for defining what it means to compress data without losing information relevant to an inferential task.

This paper builds on that idea but shifts the object of compression. Classical sufficiency begins with a specified likelihood and asks which statistic preserves information about the parameter. Representation learning often begins instead with high-dimensional observations whose relevant variation is structured by symmetries [rosebrock_visual_2024]. In this setting, the central question is not only which statistic is sufficient for a parameter, but which symmetries remove the most nuisance variation while preserving task-specific structure.

We formalize this connection by studying group actions on the sample space. If a group $G$ acts on $X$, then each orbit $Gx$ collects observations that differ only by the symmetry action. When the likelihood or conditional target law is invariant along these orbits, the quotient map $\pi:X\to X/G$ is a sufficient representation. Under an orbit-separation condition, it is minimally sufficient. Thus, the orbit space gives a geometric analogue of the minimal sufficient statistic: it is the coarsest representation that preserves the information relevant to the invariant task.

Information Geometry

Information geometry studies statistical models as manifolds whose points are probability distributions. In this view, learning takes place on a parametric manifold, and the Fisher information metric describes the local geometry of distinguishability among nearby distributions [amari_information_2016] [chirikjian_engineering_2000]. Chentsov’s theorem gives a foundational invariance result: under suitable conditions [dowty_chentsovs_2017] [halverson_naturalness_2026], the Fisher metric is the classical Riemannian structure preserved by sufficient statistic-like Markov morphisms.

Our perspective is complementary. Rather than beginning with the manifold of parametric distributions, we begin with the geometry of the sample space itself. When the sample space carries a group action, the relevant geometric operation is quotienting by the orbits of nuisance symmetries [rosebrock_visual_2024] [serre_linear_1977] [sagan_symmetric_2001]. This converts a symmetry structure on data into a candidate sufficient representation. The resulting quotient $X/G$ is not a statistical manifold of distributions, but a reduced observation space that removes nuisance structure from the representation.

This distinction is important for representation learning. In many machine learning settings, the parametric family is implicit, overparameterized, or difficult to interpret directly. The geometry of the data and the symmetries of the task may be more accessible than the geometry of the learned model’s full parameter space. Our contribution is to connect these two views. Under invariant statistical structure, geometric quotients are statistically sufficient for inference.

Geometric Deep Learning and Invariant Representations

Geometric deep learning studies learning problems with known or hypothesized symmetries. A representation is invariant if it is unchanged under a group action, and equivariant if it transforms predictably when the input is transformed [rosebrock_visual_2024]. These principles have led to architectures for images, graphs, sets, manifolds, and physical systems, where respecting symmetry can improve generalization and reduce sample complexity.

Our work differs in its emphasis. Much of geometric deep learning asks how to design architectures that respect a symmetry. We instead ask when an invariant representation is minimal sufficient for inference. It turns out that the answer is task-dependent. For invariant tasks, variation along group orbits is nuisance variation, and a minimal representation should collapse each orbit. For equivariant or reconstructive tasks, however, nontrivial group coordinates may be necessary and should not be treated as redundancy.

This distinction clarifies the role of irreducible representations. Decomposing a representation into irreducible components identifies the elementary harmonic modes through which the group acts [greiner_quantum_1989] [hall_lie_2015]. But irreducibility alone is not the same as minimal sufficiency. A nontrivial irreducible component may be redundant for an invariant classification task and essential for an equivariant prediction task. Minimality is therefore not a property of the representation alone; it is a property of the representation relative to the task at-hand.

Harmonic Analysis and Minimal Irreducible Representations

For compact groups and homogeneous spaces, harmonic analysis deconstructs the function into the modes of irreducible harmonics. The Peter-Weyl theorem decomposes square-integrable functions on a compact group into matrix coefficients of irreducible unitary representations [kondor_clebsch-gordan_2018] [donkin_clebsch-gordan_2020]. These matrix coefficients generalize ordinary Fourier modes to noncommutative groups.

This harmonic viewpoint connects directly to sufficiency for exponential families on compact groups. If a harmonic exponential family is defined by a finite set of irreducible representation coefficients [cohen_harmonic_2015] [tsipidi_harmonic_2025] [sundararajan_fourier_2018], then the corresponding Fourier coefficients are the natural sufficient statistics. Under full-rank conditions, these statistics are minimally sufficient for the model parameters. Thus, harmonic coefficients provide a finite-band statistical realization of the more general quotient-sufficiency principle.

The quotient and harmonic views play different roles. The quotient $X/G$ identifies the minimal invariant representation with respect to the sample space. The harmonic decomposition provides coordinates for representing functions, densities, and finite-band exponential families on compact groups or homogeneous spaces. Our aim is to connect these levels: orbit spaces describe what invariant representations should collapse, while irreducible harmonic coefficients describe the structure of those irreducibles.

Theoretical Results

This section develops the connection between minimal statistical sufficiency, orbit spaces, and harmonic coordinates on compact groups. We do this in four steps. First, we show that when a likelihood or prediction task is invariant under a group action, the orbit quotient is a sufficient representation. We then use standard statistical theory to define the conditions for minimal sufficiency. Second, we analyze compact Lie groups, where Peter-Weyl theory provides harmonic coordinates through irreducible unitary representations. Third, we show that for finite-band harmonic exponential families, empirical generalized Fourier coefficients are minimal sufficient statistics. Finally, we express the partition function of these families algebraically using Clebsch-Gordan decomposition.

Minimal Sufficient Invariant Quotients

Let $G$ be a compact group acting measurably on a sample space $\mathcal X$, and let

$$\pi:\mathcal X\to \mathcal X/G$$

denote the quotient map sending each point to its orbit. The quotient identifies observations that differ only by the group action.

Theorem (Sufficiency of the Orbit Quotient)

Let $\{P_\theta:\theta\in\Theta\}$ be a dominated statistical family on $\mathcal X$ with density $p_\theta(x)$. Suppose the likelihood is invariant along group orbits:

$$p_\theta(gx)=p_\theta(x) \quad \text{for all } g\in G,\ x\in\mathcal X,\ \theta\in\Theta.$$

Then $\pi(X)$ is sufficient for $\theta$.

We prove Theorem 1 in Appendix A. For relevant tasks, symmetries indicate nuisance structure in the representation that can be safely reduced. Projecting our likelihood onto the orbit space compresses the symmetries into some minimal form that preserves predictive performance.

More recent work in representations seek to define the minimal structure that preserves task-relevant information. This minimal sufficient representation is analogous to achieving the information-theoretic lower bound using the coarsest possible representation, and effectively balances the precision of the representation with its accuracy.

Theorem (Minimal Sufficient Orbit Quotient)

Assume, in addition, that distinct orbits are statistically distinguishable:

$$\pi(x)\neq \pi(x') \quad\Longrightarrow\quad \frac{p_\theta(x)}{p_\theta(x')} \text{ is not constant in }\theta.$$

Then $\pi(X)$ is minimal sufficient for $\theta$.

ProofShow

This result formalizes the sense in which an orbit space is a minimal invariant representation. If the task cannot distinguish points within the same orbit, then orbit-level information is sufficient. If distinct orbits remain statistically distinguishable, then no coarser invariant representation is sufficient.

It also uses the likelihood ratio as a criterion, which is that classical method to prove minimality of a sufficient statistic [casella_statistical_2024]. As such, it demonstrates the relevance of statistical minimality to the wider field of representation theory and abstract algebraic groups.

Generalized Fourier Statistics on Compact Groups

We define the sample on a compact Lie group $G$ with normalized Haar measure $dg$. Let $X_1,\dots,X_n$ be i.i.d. random variables taking values in $G$, with density $p\in L^2(G)$.

Let $\widehat G$ denote the set of equivalence classes of irreducible unitary representations of $G$. For $\lambda\in\widehat G$, let

$$\rho_\lambda:G\to U(d_\lambda)$$

be an irreducible unitary representation of dimension $d_\lambda$. The generalized Fourier coefficient of $p$ at frequency $\lambda$ is

$$\widehat p(\lambda) = \int_G p(g)\rho_\lambda(g)\,dg.$$

Given observations $X_1,\dots,X_n$, its empirical estimate is

$$\widehat p_{\mathrm{emp}}(\lambda) = \frac{1}{n}\sum_{i=1}^n \rho_\lambda(X_i).$$

For a finite subset $\widehat G_N\subset\widehat G$, define the Peter-Weyl projection

$$(\mathcal T_Np)(g) = \sum_{\lambda\in\widehat G_N} d_\lambda \operatorname{Tr}\left( \widehat p(\lambda)\rho_\lambda(g) \right).$$

This is a finite-band approximation of $p$, retaining only the harmonic components indexed by $\widehat G_N$.

Minimal Sufficiency of Finite-Band Harmonic Statistics

Consider the finite-band harmonic exponential family

$$p_\theta(g) = \exp\left[ E_\theta(g)-A(\theta) \right],$$

where

$$E_\theta(g) = \sum_{\lambda\in\widehat G_N} d_\lambda \operatorname{Tr} \left( C_\lambda\rho_\lambda(g) \right),$$

and $\theta=\{C_\lambda\}_{\lambda\in\widehat G_N}$.

Theorem (Minimal Sufficiency of Harmonic Statistics)

Suppose the finite-band harmonic exponential family above is full rank: after choosing real coordinates for the matrix coefficients, the natural statistic has no nontrivial affine dependence almost surely, and the natural parameter space contains an open set. Then, for i.i.d. observations $X_1,\dots,X_n\in G$, the statistic

$$T_N(X_1,\dots,X_n) = \left( \sum_{i=1}^n \rho_\lambda(X_i) \right)_{\lambda\in\widehat G_N}$$

is minimal sufficient for $\theta$.

ProofShow

Algebraic Normalization by Clebsch-Gordan Decomposition

The partition function is

$$e^{A(\theta)} = \int_G e^{E_\theta(g)}dg.$$

Although $E_\theta$ is finite-band, $e^{E_\theta}$ generally is not. This makes it intractable both to compute and analyze algebraically. The following result defines the normalization constant algebraically.

Theorem (Clebsch-Gordan Normalization)

For the finite-band harmonic exponential family above,

$$e^{A(\theta)} = \sum_{k=0}^{\infty} \frac{1}{k!} \operatorname{Coeff}_{\mathbf 1} \left(E_\theta^k\right),$$

where $\operatorname{Coeff}_{\mathbf 1}(E_\theta^k)$ is the coefficient of the trivial representation in the Peter-Weyl expansion of $E_\theta(g)^k$, obtained by decomposing tensor products of irreducible representations using Clebsch-Gordan rules.

ProofShow

This theorem swaps continuous integration over $G$ with a combinatorial sum. Though the infinite sum is intractable to compute, it is algebraically interpretable when compared with integrating the energy function directly.

Peter-Weyl Recovery

Theorem (Harmonic Completeness)

Let $p\in L^2(G)$. The algebraic span of the matrix coefficients of irreducible representations of $G$ is dense in $L^2(G)$. Therefore, for any sequence of finite subsets $\widehat G_N$,

$$\lim_{N\to\infty} \|p-T_Np\|_{L^2(G)} = 0.$$

This result establishes harmonic completeness. It shows that no $L^2$ information is lost in the infinite-band limit. The sufficiency result above is instead a finite-band statement about a particular exponential family.

Abstracting to Task-Complete Representation

We now define a complete representation analogous to that of classical Fourier completeness. This requirement is more stringent than minimal sufficiency, requiring sufficient inference over a class of tasks $\mathcal F.$

Definition (Task-Complete Representation)

Let $\mathcal F$ be a family of tasks. Each task induces a target law or likelihood:

$$K_f(x) = p_f(y|x).$$

If we define an equivalence relation, $x \sim x' \iff K_f(\cdot \mid x) = K_f(\cdot \mid x'),$ then the minimal sufficient representation $q_{\mathcal F}$ is sufficient for the family of tasks such that $\mathcal X \to \mathcal X / {\sim}$.

For a single invariant task, the quotient group is the minimal sufficient representation. As we expand the list of evaluation tasks, the quotient group refines progressively, indicating the distinct group actions under which a representation is invariant.

Corollary (Compactness of Multi-Task Invariance Subgroups)

Let $G$ be a compact Lie group and let $G_f \subseteq G$ be the closed invariance subgroup for each task $f \in \mathcal{F}$. Then the joint invariance group $G_{\mathcal{F}} = \bigcap_{f \in \mathcal{F}} G_f$ is a closed, compact subgroup of $G$. Consequently, the multi-task quotient space $\mathcal{X} / G_{\mathcal{F}}$ permits a canonical Peter-Weyl harmonic decomposition.

The above follows from canonical group theory. The intersection of compact Lie groups is also a compact Lie group [hall_lie_2015]. This enables the Clebsch-Gordan decomposition of the multi-task irreducible representations.

Discussion

This work develops a geometric–statistical account of minimal sufficient representations for learning tasks with group-structured symmetries. By treating the quotient X/G as a statistic of the data and applying classical sufficiency theory, we showed that group orbits encode exactly the nuisance variation that can be collapsed without sacrificing inferential power for invariant tasks. Under an orbit-separation condition, this quotient becomes minimal in the precise sense that no coarser invariant representation remains sufficient, connecting representation learning directly to the classical likelihood-ratio characterization of minimal sufficient statistics.

Whereas information geometry begins with a manifold of probability distributions endowed with the Fisher metric, our starting point is the geometry of the sample space and its symmetry group. Geometric deep learning typically asks how to construct architectures that are invariant or equivariant to a given group, often working in generalized Fourier coordinates but without explicitly characterizing when those coordinates are minimal for a task. Our results instead characterize when the symmetry quotient is sufficient and minimally sufficient, and they clarify that minimality is inherently task-relative: an irreducible harmonic component may be redundant for an invariant classification task yet indispensable for an equivariant or reconstructive task.

Beyond single tasks, we introduced task-complete representations based on a family of tasks $\mathcal F$. Each task induces a closed invariance subgroup $G_f$, and their intersection $G_{\mathcal F}$ is itself a compact Lie subgroup that has a Peter-Weyl decomposition into its harmonic irreducibles. The quotient $X/G_{\mathcal F}$ thus defines a representation that is simultaneously invariant, and under suitable separation conditions minimally sufficient, for the entire task family, providing a structural analogue to algorithmic approaches such as Minimal Achievable Sufficient Statistic learning. This multi-task perspective is particularly relevant for AI safety and interpretability, where representations must support a variety of downstream tasks without encoding unnecessary complexity.

At the same time, our analysis rests on several idealizations. We assume dominated statistical families with exact invariance and orbit-separation, whereas practical models often exhibit approximate invariance and are specified only implicitly through neural network parameterizations. Extending the theory to approximate sufficiency and robustness to model mis-specification is a natural direction for future work. Similarly, our harmonic results focus on compact Lie groups and finite-band expansions; many physical and social systems involve non-compact or approximate symmetry groups for which Peter–Weyl theory and Clebsch–Gordan rules require careful adaptation. Finally, while we outlined the structural properties of minimal sufficient quotients, translating these into practical learning objectives and architectures—potentially integrating MASS-style losses or group-equivariant layers—is an open engineering and empirical question.

Taken together, the results here suggest a view of representation learning in which minimality is determined not only by compression or mutual information but by the geometry of group actions and the sufficiency of quotient statistics. By unifying orbit quotients, harmonic coordinates, and exponential-family sufficiency, this framework offers a principled target for the design and evaluation of representations in settings where symmetries, task families, and safety constraints are all central concerns.

Appendix A: Proof of the Sufficiency of the Orbit Quotient

ProofShow

Appendix B: Truncation Bounds on the Clebsch-Gordan Decomposition

The summands of the Clebsch-Gordan decomposition are technically infinite, which is intractable to efficiently compute. Depending on the level at which we truncate, $k,$ there is a tradeoff between precision and computability.

Theorem (Truncation Bound)

$$\left| e^{A(\theta)} - \sum_{k=0}^{K} \frac{1}{k!}W_{0,k}(\theta)\right| \leq R_K(\|\theta\|).$$

ProofShow

Appendix C: Proof of Compactness of Multi-Task Minimal Sufficient Subgroups

We establish the corollary in four steps.

1. By hypothesis, each task invariance group $G_f$ is a closed subset of $G$. In point-set topology, the intersection of an arbitrary family of closed sets is closed. Therefore,

$$G_{\mathcal{F}} = \bigcap_{f \in \mathcal{F}} G_f$$

is a closed subset of $G$. Since $G$ is a compact space, any closed subset of $G$ is automatically compact. Thus, $G_{\mathcal{F}}$ is compact. 2. To show $G_{\mathcal{F}}$ is a subgroup, let $g, h \in G_{\mathcal{F}}$. By definition of an intersection, $g, h \in G_f$ for every $f \in \mathcal{F}$. Because each $G_f$ is a subgroup, the product $gh^{-1} \in G_f$ for every $f \in \mathcal{F}$. Therefore, $gh^{-1} \in \bigcap_{f \in \mathcal{F}} G_f = G_{\mathcal{F}}$, satisfying the subgroup criterion. 3. By Cartan’s theorem (the Closed Subgroup Theorem) [sagan_symmetric_2001] [rosebrock_visual_2024] [serre_linear_1977], any topologically closed subgroup of a Lie group is an embedded Lie subgroup. Since $G_{\mathcal{F}}$ is a closed subgroup of the Lie group $G$, it inherits a unique smooth manifold structure making it a compact Lie subgroup. 4. Because $G_{\mathcal{F}}$ is a compact Lie group, the classical Peter–Weyl theorem guarantees that the algebraic span of its matrix coefficients is dense in $L^2(G_{\mathcal{F}})$. Furthermore, the transition from a single task’s invariance group $G_f$ to the multi-task subgroup $G_{\mathcal{F}}$ can be understood geometrically as symmetry breaking. By the theory of branching laws, the restriction of any irreducible unitary representation of $G_f$ to the compact subgroup $G_{\mathcal{F}}$ decomposes discretely into a direct sum of irreducible representations of $G_{\mathcal{F}}$.