Branch-Symmetric Nonidentifiability

• Branch-symmetric nonidentifiability is an umbrella concept where symmetry induces multiple equivalent parameter branches, making them observationally indistinguishable.
• It differentiates between intrinsic model symmetries (e.g., latent variable transformations) and representation ambiguities (e.g., repeated eigenvalue bases), each affecting identifiability uniquely.
• Applications span latent graphs, tensor decompositions, ODE/PDE models, and skew-normal frameworks, illustrating both theoretical challenges and limits of empirical identifiability.

“Branch-symmetric nonidentifiability” (Editor's term) is not a standard label in the cited literature. The phrase most naturally denotes situations in which several symmetry-related branches of parameters, decompositions, eigenspace coordinates, latent representations, or solution families are observationally equivalent, generate the same input-output behavior, or remain empirically indistinguishable. Across the literature, the closest established notions include model-based nonidentifiability and subspace nonidentifiability in latent position random graphs, field-dependent failure of tensor identifiability over $\mathbb C$ versus $\mathbb R$, non-identifiability under permutation of latent variables in unified skew-normal models, and finite or continuous parameter fibers in ODE models (Agterberg et al., 2020, Angelini et al., 2016, Wang et al., 2023, Ovchinnikov et al., 2023).

1. Terminological scope and principal distinctions

The literature separates several phenomena that a branch-oriented vocabulary can conflate. In latent position random graph models, one source of ambiguity comes from transformations to the inputs under which $\kappa$ is invariant; another comes from the nonuniqueness of eigenvector bases for repeated eigenvalues. The former is model-based nonidentifiability and the latter is subspace nonidentifiability (Agterberg et al., 2020). In spectral optimization, identifiability is transferred from eigenvalue space to matrix space only after replacing branchwise descriptions by invariant manifolds of the form $\lambda^{-1}(M)$ (Daniilidis et al., 2013). In analytic function theory, symmetric observables built from the full branch set of the Lambert $W$-function lose branch labels by construction (Cohen, 2020).

This suggests that branch-symmetric nonidentifiability is best treated as an umbrella notion rather than a single formal object. In some settings the ambiguity is a genuine many-to-one parameter map; in others it is a basis-representation problem; in still others it is a consequence of observing only a symmetry-invariant summary.

Domain	Symmetry or branch mechanism	Closest formal notion
Latent position random graphs	$X \mapsto XQ$, repeated-eigenvalue basis changes	Model-based / subspace nonidentifiability
Tensor decomposition	Multiple secant decompositions, real versus complex branches	Failure of identifiability over $\mathbb C$ or $\mathbb R$
Unified skew-normal models	Permutation of latent truncation coordinates	Non-identifiability under permutation of latent variables
Spectral matrix optimization	Orthogonal orbits, repeated eigenvalues	Identifiability / partial smoothness of $\lambda^{-1}(M)$
ODE models	Fibers over identifiable parameter combinations	Local IO-identifiability
Nonlinear elliptic PDEs	Symmetric and non-symmetric bifurcation branches	Multiplicity of critical branches

A recurrent technical distinction is between intrinsic symmetry of the model and symmetry of a representation. The first changes parameters without changing the induced law or input-output equations. The second changes coordinates, bases, or eigenvectors without changing the invariant object being estimated. That distinction is central throughout the subject (Agterberg et al., 2020, Daniilidis et al., 2013).

2. Symmetry groups, invariant actions, and branch-invariant observables

In latent position random graphs with kernel $\kappa$, model-based nonidentifiability arises whenever there exists a transformation $\mathbb R$0 such that

$\mathbb R$1

for all $\mathbb R$2. In the generalized random dot product graph,

$\mathbb R$3

and observationally equivalent latent configurations arise under any $\mathbb R$4 satisfying

$\mathbb R$5

that is, $\mathbb R$6. In the RDPG special case $\mathbb R$7, this reduces to $\mathbb R$8 (Agterberg et al., 2020).

In unified skew-normal models, the symmetry is discrete rather than orthogonal. If $\mathbb R$9 is an $\kappa$0 permutation matrix, then

$\kappa$1

produce the same observed law for $\kappa$2. The paper proves that

$\kappa$3

if and only if $\kappa$4, and concludes that the SUN family is non-identifiable if $\kappa$5 (Wang et al., 2023).

In orthogonally invariant matrix optimization, the relevant symmetry is encoded by stabilizers. For a point $\kappa$6, the stabilizer is

$\kappa$7

while for $\kappa$8,

$\kappa$9

The subdifferential formula

$\lambda^{-1}(M)$0

shows that branchwise eigenvector descriptions are subordinate to an orthogonal orbit. The identifiable object is therefore not a single eigenbasis branch but the lifted manifold $\lambda^{-1}(M)$1 (Daniilidis et al., 2013).

An analytic archetype appears for Lambert $\lambda^{-1}(M)$2. The paper proves the exact symmetric identity

$\lambda^{-1}(M)$3

together with

$\lambda^{-1}(M)$4

These formulas show that symmetric aggregation over all branches can erase branch labels completely, and in the second identity even the dependence on $\lambda^{-1}(M)$5 disappears (Cohen, 2020).

3. Tensor decomposition, secant geometry, and field-dependent branches

Tensor decomposition provides the most explicit branch language in the cited literature. A decomposition of length $\lambda^{-1}(M)$6 is encoded by the abstract secant incidence

$\lambda^{-1}(M)$7

with projection

$\lambda^{-1}(M)$8

Identifiability means that the fiber $\lambda^{-1}(M)$9 is essentially one point modulo reorderings and rescalings; nonidentifiability means that the fiber has multiple points or positive dimension (Angelini et al., 2016).

The branch interpretation is particularly sharp over $W$0 and $W$1. The paper on real versus complex identifiability studies cases where

$W$2

For a real elliptic normal curve $W$3 of degree $W$4, there are real points lying in the intersection of two $W$5-spaces $W$6 such that one secant branch meets $W$7 in $W$8 real points while the other meets $W$9 in $X \mapsto XQ$0 points, some non-real; moreover there exists a nonempty Euclidean-open subset of points with the same property (Angelini et al., 2016).

The same paper distinguishes finite branch multiplicity from positive-dimensional nonuniqueness. In the finite case, a Euclidean-open set may contain tensors with exactly one real decomposition and several non-real ones. In the positive-dimensional case, the paper shows that if a real decomposition exists generically, then there are infinitely many real decompositions as well, so real identifiability cannot be recovered on a nontrivial Euclidean-open set (Angelini et al., 2016).

The symmetric examples are especially explicit. For ternary septics,

$X \mapsto XQ$1

a general complex tensor has rank $X \mapsto XQ$2 and exactly $X \mapsto XQ$3 decompositions, while the paper constructs a nontrivial Euclidean-open subset of real tensors with exactly one real decomposition and $X \mapsto XQ$4 non-real ones. For ternary octics,

$X \mapsto XQ$5

a general tensor has rank $X \mapsto XQ$6 and exactly $X \mapsto XQ$7 decompositions, and the paper finds a Euclidean-open subset of real tensors with exactly one real decomposition and $X \mapsto XQ$8 non-real decompositions (Angelini et al., 2016).

A complementary birational-geometric mechanism appears for partially symmetric tensors. For an irreducible projective variety $X \mapsto XQ$9, the tangential $\mathbb C$0-projection

$\mathbb C$1

controls generic identifiability. The paper proves that under Mori fiber space and nefness assumptions, if $\mathbb C$2 is perfect, nondefective, and not $\mathbb C$3-twd, then $\mathbb C$4 is not birational; hence $\mathbb C$5 is not generically identifiable. Applied to Segre–Veronese varieties, this yields infinite families of partially symmetric tensor spaces with generic nonuniqueness of rank-$\mathbb C$6 decompositions (Casarotti et al., 2020).

4. Spectral embeddings, latent representations, and coordinate artifacts

In latent position random graphs, the base model is

$\mathbb C$7

with generalized Gram matrix

$\mathbb C$8

The paper’s central point is that two distinct sources of nonidentifiability arise naturally here. Model-based nonidentifiability is induced by invariances of $\mathbb C$9, whereas subspace nonidentifiability is induced by nonuniqueness of eigendecomposition bases when eigenvalues repeat (Agterberg et al., 2020).

The spectral formulation makes the distinction concrete. Let

$\mathbb R$0

be the adjacency spectral embedding, and write the spectral factorization of $\mathbb R$1 as

$\mathbb R$2

The matrix $\mathbb R$3 is defined by

$\mathbb R$4

If $\mathbb R$5 and $\mathbb R$6 is full rank with distinct eigenvalues, then there exists a deterministic indefinite orthogonal matrix $\mathbb R$7 such that

$\mathbb R$8

If repeated eigenvalues are present, then there exist matrices

$\mathbb R$9

and a deterministic $\lambda^{-1}(M)$0 such that

$\lambda^{-1}(M)$1

Thus model-based ambiguity becomes asymptotically controllable, whereas subspace ambiguity remains tied to block-orthogonal freedom inside repeated-eigenvalue subspaces (Agterberg et al., 2020).

Orthogonally invariant optimization formulates the same issue at the level of active manifolds. The paper proves that if $\lambda^{-1}(M)$2 is locally symmetric around $\lambda^{-1}(M)$3, then

$\lambda^{-1}(M)$4

It also proves

$\lambda^{-1}(M)$5

This treats many apparent branch ambiguities from repeated eigenvalues as artifacts of using a non-invariant description (Daniilidis et al., 2013).

A counterexample from unsupervised representation learning shows that nonidentifiability need not come from a finite group. The construction takes

$\lambda^{-1}(M)$6

and defines an alternative representation

$\lambda^{-1}(M)$7

where

$\lambda^{-1}(M)$8

The two representations generate the same $\lambda^{-1}(M)$9-algebra, both have compact support, and both satisfy the independent support condition, yet $\kappa$0 cannot in general be expressed as a permutation and coordinate-wise bijection transformation. This is a piecewise latent automorphism rather than a discrete branch exchange, but it serves the same role: observational indistinguishability with nontrivial latent reparameterization (Ghosh et al., 2022).

5. Statistical models, empirical distinguishability, and the limits of observation

The unified skew-normal family gives a direct instance of symmetry-induced nonidentifiability. Starting from the latent Gaussian representation

$\kappa$1

the paper proves that permutation of the latent truncation coordinates leaves the observed law unchanged. The transformation

$\kappa$2

for $\kappa$3 generates equivalent parameter branches, and the SUN family is non-identifiable if $\kappa$4 (Wang et al., 2023).

The same paper embeds the phenomenon into a broader class of selection models. If $\kappa$5, where $\kappa$6 is positive diagonal and $\kappa$7 a permutation matrix, then order-preserving transformations of the latent selection coordinates preserve the relevant selection events. For SUN, the correlation-matrix normalization removes the diagonal-scaling freedom but leaves the permutation symmetry intact (Wang et al., 2023).

A different limitation arises when a parameter is classically identifiable but not empirically identifiable. The paper on empirical identifiability defines $\kappa$8 to be empirically identifiable if there exists a consistent sequence of estimators

$\kappa$9

It then proves that in the Gaussian constant-correlation model, the correlation parameter $\mathbb R$00 is not empirically identifiable, the mean $\mathbb R$01 is not empirically identifiable, and two values $\mathbb R$02 with $\mathbb R$03 are not even potentially distinguishable in any direction. In the fixed classification model for $\mathbb R$04-means, the parameters $\mathbb R$05 are also not empirically identifiable (Hennig, 2021).

This observational perspective sharpens the branch language. Distinct parameter branches may induce different full laws, yet the available asymptotic regime may never accumulate enough information to separate them uniformly over nuisance fibers. A branch ambiguity can therefore be exact at the model level, field-dependent, or purely empirical (Hennig, 2021).

Phylogenetic branch lengths supply an instructive contrast. Under the symmetric $\mathbb R$06-state model, the map from branch lengths $\mathbb R$07 to tree split probabilities is proved locally invertible near $\mathbb R$08 for arbitrary trees with no degree-$\mathbb R$09 internal vertices, and globally one-to-one for trees with at most four leaves. The general global case is left open, and the paper gives no finite-branch-length counterexample to identifiability. The only explicit degeneracy occurs in the $\mathbb R$10-parameterization at $\mathbb R$11, corresponding to $\mathbb R$12 (Chor et al., 2013).

6. ODE reparameterization and symmetry-breaking branches in PDEs

For rational ODE models

$\mathbb R$13

the paper on identifiable specializations proves a universal existence theorem: there exists a tuple

$\mathbb R$14

of the same length as $\mathbb R$15 such that the entries of $\mathbb R$16 are locally IO-identifiable and $\mathbb R$17 has the same input-output equations as $\mathbb R$18. If the sum of the orders of the IO-equations equals the state dimension $\mathbb R$19, then the state variables of $\mathbb R$20 can be expressed as algebraic functions of $\mathbb R$21 and $\mathbb R$22 (Ovchinnikov et al., 2023).

The constructive specialization step preserves the identifiable invariants $\mathbb R$23 by solving

$\mathbb R$24

subject to rank-preserving nonvanishing conditions. This replaces a nonidentifiable parameter fiber by a locally identifiable representative while keeping the same shape of the equations: the monomials in the new state variables are formed in the same way as in the original model (Ovchinnikov et al., 2023).

The examples separate discrete and continuous branch structures. In

$\mathbb R$25

the IO-equation

$\mathbb R$26

shows that only $\mathbb R$27 and $\mathbb R$28 are globally identifiable, so $\mathbb R$29 and $\mathbb R$30 remain only locally identifiable as roots of a quadratic; this is a permutation branch. In the harmonic oscillator

$\mathbb R$31

only the product $\mathbb R$32 is identifiable, so the fiber is positive-dimensional and the method fixes a representative such as $\mathbb R$33. In Lotka–Volterra, the nonidentifiable parameter $\mathbb R$34 can be specialized to $\mathbb R$35 and absorbed into a state rescaling (Ovchinnikov et al., 2023).

Nonlinear elliptic PDEs exhibit a related but distinct use of branch language. For extremals of Caffarelli–Kohn–Nirenberg inequalities, the paper studies bifurcation of non-symmetric critical points from the explicit symmetric branch. The linearized operator has lowest eigenvalue

$\mathbb R$36

and the first bifurcation point is

$\mathbb R$37

Beyond this threshold, non-symmetric branches appear. For some parameter regimes,

$\mathbb R$38

so non-symmetric solutions exist for values of $\mathbb R$39 smaller than the bifurcation value. The paper also reports that when $\mathbb R$40, symmetric and non-symmetric optimal functions coexist (Dolbeault et al., 2013).

This is not nonidentifiability in the statistical sense. It is multiplicity of critical branches created by symmetry and symmetry breaking. Even so, it is structurally close to branch-symmetric nonidentifiability because the same control parameter can correspond to several distinct symmetry-related or symmetry-broken solutions (Dolbeault et al., 2013).

7. Synthesis, common misconceptions, and invariant targets

Taken together, these works suggest that branch-symmetric nonidentifiability should be analyzed at three levels. First, some ambiguities are induced by model symmetries: $\mathbb R$41 in GRDPG, latent-coordinate permutations in SUN, or parameter fibers defined by identifiable combinations in ODE models (Agterberg et al., 2020, Wang et al., 2023, Ovchinnikov et al., 2023). Second, some are induced by representational choices: repeated-eigenvalue basis changes, orthogonal orbit descriptions, or symmetric observables of multivalued analytic branches (Daniilidis et al., 2013, Cohen, 2020). Third, some are induced by the field or observational regime: real versus complex decompositions of the same tensor, or parameters that are identifiable in principle but not empirically identifiable from the available asymptotics (Angelini et al., 2016, Hennig, 2021).

A common misconception is to treat all branch multiplicities as the same phenomenon. The cited literature does not support that. A finite symmetry orbit, a positive-dimensional secant fiber, a repeated-eigenvalue eigenspace, and a left-bending bifurcation branch have different mathematical status and different inferential consequences. The latent graph literature explicitly separates model-based nonidentifiability from subspace nonidentifiability, and tensor geometry distinguishes finite decomposition multiplicity from positive-dimensional nonuniqueness (Agterberg et al., 2020, Angelini et al., 2016).

Another misconception is that symmetry necessarily implies failure of meaningful inference. Often the correct inferential target is a quotient or invariant object. In graph embedding, the asymptotic target may be $\mathbb R$42 or the deterministic transform $\mathbb R$43, not the original latent positions. In orthogonally invariant optimization, the identifiable object is the symmetry-saturated manifold $\mathbb R$44. In SUN models, one may restore identifiability by imposing ordering constraints on $\mathbb R$45 or on the eigenvalues of $\mathbb R$46. In ODE models, partial specialization can produce a locally identifiable model with the same input-output equations (Agterberg et al., 2020, Daniilidis et al., 2013, Wang et al., 2023, Ovchinnikov et al., 2023).

A final caution is that global elimination of branches is not always possible. The ODE specialization theorem guarantees only local IO-identifiability, and the paper gives explicit examples where no globally identifiable reparametrization exists. Likewise, tensor identifiability can fail generically even when the tangential contact locus is zero-dimensional, and the general global identifiability question for phylogenetic branch lengths from split probabilities remains open rather than negatively resolved (Ovchinnikov et al., 2023, Casarotti et al., 2020, Chor et al., 2013).

Under this synthesis, branch-symmetric nonidentifiability is best understood not as a single theorem but as a recurrent structural pattern: symmetry organizes solution sets into branches, and identifiability depends on whether one seeks a single representative, an invariant quotient, a field-specific decomposition, or an empirically recoverable target.