Generalized Similarities: Theory & Applications

• Generalized similarities are a family of constructs that relax traditional similarity metrics to include varied transformations, indices, and relations.
• They are applied in domains ranging from cut-and-project geometry and cognitive science to non-Hermitian physics and graph databases.
• Key methodologies include linear maps, concave distance-decay laws, and similarity transition operators that enhance robust learning and structural analysis.

Searching arXiv for the provided topic and papers to ground the article in current records. Generalized similarities are extensions of classical similarity notions in which comparison is no longer restricted to Euclidean shape similarity or to fixed feature overlap. In recent research, the term denotes several formally distinct constructions: linear maps $A$ on $\mathbb{R}^n$ satisfying $A\Sigma\subset\Sigma$ for cut-and-project sets; concave distance–decay laws $g(d)\approx a e^{-bd}+c$ in psychological space; Kronecker/Jaccard-derived similarity indices on scalars, vectors, and functions; qualitative relations induced by shared term-generalizations in universal algebra; pseudo-Hermitian and skew-similar constraints on non-Hermitian Hamiltonians; structurally robust similarities on graph databases; and similarity-transition operators between base and novel classes in generalized few-shot segmentation (Masáková et al., 2019, Marjieh et al., 2023, Costa, 2021, Antić, 2023, Montag et al., 2024, Wang et al., 2024).

1. Scope and recurring formal pattern

Across the literature, generalized similarity is not a single invariant but a family of constructions that enlarge what counts as “the same,” “close,” or “compatible.” In some settings it is a scalar score; in others it is a preorder, a lattice-theoretic relation, a symmetry constraint, or a transformation law. A recurrent pattern is that a classical notion is relaxed by enlarging the admissible transformations, supports, or latent spaces. This suggests a common meta-structure: generalized similarities preserve a chosen form of organization while relaxing the ambient representation.

Domain	Object	Defining relation
Cut-and-project geometry	self-similarity of $\Sigma$	$A\Sigma\subset\Sigma$
Cognitive generalization	similarity vs. distance	$g(x)=a e^{-bx}+c$
Scalar/vector/function similarity	yielding indices	$s_1,s_2,s_3,s_4$
Universal algebra	similarity by generalizations	$a\approx b$ via $\uparrow_\mathfrak{A}a$
Non-Hermitian physics	generalized similarities of $\mathbb{R}^n$0	$\mathbb{R}^n$1, $\mathbb{R}^n$2, $\mathbb{R}^n$3
GFSS	class similarity transition	$\mathbb{R}^n$4

A common misconception is to treat generalized similarity as necessarily metric or necessarily symmetric. The cited work explicitly includes asymmetric probabilistic transitions, inclusion relations such as $\mathbb{R}^n$5, and reflexive symmetric but non-transitive relations defined by maximal overlap of generalizations (Antić, 2023, Wang et al., 2024).

2. Geometric, quasicrystalline, and fractal self-similarities

In the theory of cut-and-project sets, generalized self-similarity means a linear map $\mathbb{R}^n$6 such that $\mathbb{R}^n$7. For diagonalizable $\mathbb{R}^n$8, existence on the scheme level is characterized exactly by the requirement that every eigenvalue of $\mathbb{R}^n$9 is an algebraic integer; for actual cut-and-project sets $A\Sigma\subset\Sigma$0, one additionally needs a bounded invariant window, yielding the sharper criterion that expanding algebraic conjugates must satisfy a multiplicity balance ensuring an internal companion $A\Sigma\subset\Sigma$1 with spectral radius $A\Sigma\subset\Sigma$2. The same framework yields explicit minimal lattice-dimension formulae in terms of irreducible factors of the minimal polynomial and their multiplicities, so the admissible generalized similarities are controlled by algebraic number theory rather than by Euclidean similarity alone (Masáková et al., 2019).

A related finite-structure phenomenon appears for iterated function systems of similarities on $A\Sigma\subset\Sigma$3. There, the generalized finite type condition is expressed through the finiteness of the overlap maps

$A\Sigma\subset\Sigma$4

so generalized similarities are the relative-position maps $A\Sigma\subset\Sigma$5. For self-similar sets that are intervals, the weak separation condition is equivalent to the finite neighbour condition and to convex generalized finite type, meaning that bounded overlap multiplicity forces only finitely many generalized similarity types of local configurations (Hare et al., 2020).

A different geometric generalization replaces a constant similarity ratio by an angle-dependent one. In Euclidean space, using the proportion $A\Sigma\subset\Sigma$6 and the golden-pair condition $A\Sigma\subset\Sigma$7, the generalized golden ratio $A\Sigma\subset\Sigma$8 is the positive root of

$A\Sigma\subset\Sigma$9

Hence $g(d)\approx a e^{-bd}+c$0 depends on the angle between vectors. The special cases $g(d)\approx a e^{-bd}+c$1, $g(d)\approx a e^{-bd}+c$2, and $g(d)\approx a e^{-bd}+c$3 recover $g(d)\approx a e^{-bd}+c$4, $g(d)\approx a e^{-bd}+c$5, and $g(d)\approx a e^{-bd}+c$6, respectively. The associated similarity set of a vector is the field

$g(d)\approx a e^{-bd}+c$7

and the same construction is extended to triangles represented as vectors in $g(d)\approx a e^{-bd}+c$8 (Grigoryan et al., 2023).

3. Distance–decay laws, overlap indices, and transform-induced similarities

In cognitive science, Shepard’s universal law of generalization states that perceived similarity should decay as a concave function of psychological distance in an appropriate psychological space. In the high-dimensional natural-image regime, this is operationalized by embedding stimuli into a 4D Euclidean MDS space and fitting

$g(d)\approx a e^{-bd}+c$9

with $\Sigma$0. The same functional form accounts for both direct similarity judgments and blank-property generalization judgments, supporting the view that similarity and inductive generalization are manifestations of a common underlying metric structure (Marjieh et al., 2023).

A more axiomatic route starts from the Kronecker delta as the prototype of strict identity and then relaxes it into continuous “yielding indices.” For scalars, the first three normalized indices are

$\Sigma$1

with $\Sigma$2 as the unbounded product form. Their signed versions take values in $\Sigma$3 for $\Sigma$4, and the construction extends systematically to multisets, vectors, and functions. In this framework, the generalized Jaccard index $\Sigma$5 is exactly $\Sigma$6, so Jaccard is interpreted as a yielding implementation of the Kronecker delta (Costa, 2021).

Random-walk–based node similarities provide another generalized metric family on graphs. For an undirected weighted graph with Laplacian $\Sigma$7, effective resistance is

$\Sigma$8

while personalized PageRank takes the form

$\Sigma$9

A full resistance matrix determines the graph Laplacian through

$A\Sigma\subset\Sigma$0

and partial noisy resistance measurements can still support substantial edge reconstruction via Laplacian optimization (Hoskins et al., 2018).

Wavelet-based image analysis shows that useful generalized similarities need not be learned. A 2D discrete wavelet transform yields a feature map $A\Sigma\subset\Sigma$1, and similarities can then be computed either through Euclidean geometry in coefficient space or via a similarity matrix $A\Sigma\subset\Sigma$2, with $A\Sigma\subset\Sigma$3 instantiated by SSIM. This permits identification of similar, redundant, and influential images before training a model, and the reported results corroborate patterns previously obtained with pre-trained CNN representations (Yousefzadeh, 2020).

4. Universal-algebraic, logical, and analogical formulations

In universal algebra, generalized similarity is defined through sets of term-generalizations rather than distances. For an algebra $A\Sigma\subset\Sigma$4 and element $A\Sigma\subset\Sigma$5,

$A\Sigma\subset\Sigma$6

collects all terms that can produce $A\Sigma\subset\Sigma$7. For $A\Sigma\subset\Sigma$8 and $A\Sigma\subset\Sigma$9, similarity is built from the overlap $g(x)=a e^{-bx}+c$0, and $g(x)=a e^{-bx}+c$1 holds when each is among the best matches for the other in terms of maximal non-trivial joint generalizations. The resulting relation is reflexive and symmetric but, in general, not transitive; it is preserved by isomorphisms, can be represented by regular tree languages in finite algebras, and specializes to modular congruence, divisibility, Green’s relations, and group conjugacy when appropriate fragments of the term language are chosen (Antić, 2023).

This algebraic notion is lifted to analogical proportions by moving from elements to arrows. In the arrow algebra $g(x)=a e^{-bx}+c$2, analogical proportion is defined by mutual similarity of transformations: $g(x)=a e^{-bx}+c$3 The resulting proportion relation satisfies p-reflexivity, p-symmetry, inner p-symmetry, and p-determinism, but in general fails inner p-reflexivity, p-transitivity, and related stronger axioms. This makes the dependence of analogy on generalized similarity explicit (Antić, 2024).

A different algebraic generalization appears in fuzzy signatures. There, a similarity relation

$g(x)=a e^{-bx}+c$4

on functor symbols is lifted to terms, first in the equal-arity setting and then in the fully fuzzy setting where similar functors may have different arities and argument orderings. The notation

$g(x)=a e^{-bx}+c$5

records both a similarity degree $g(x)=a e^{-bx}+c$6 and an injective position mapping $g(x)=a e^{-bx}+c$7. This supports weak unification and generalization procedures that tolerate mismatches on functor names, arity, and argument order while preserving conventional data structures for terms and substitutions (Aït-Kaci et al., 2017).

The universal-algebraic study of term generalization modulo equational theories recasts similarity through generality posets of solutions. Problems have unitary, finitary, infinitary, or nullary type depending on the cardinality of minimal complete sets of least general solutions. The key structural result is that, in 1ESP varieties, the solution poset is dually isomorphic to the poset of projective congruences below the problem kernel in the congruence lattice of the 1-generated free algebra. Abelian groups, commutative monoids, commutative semigroups, Boolean algebras, Gödel algebras, Kleene algebras, lattices, semilattices, and idempotent varieties without constants are all exhibited as unitary examples (Flaminio et al., 25 Feb 2025).

5. Structural invariance, spectral symmetry, and representation independence

In non-Hermitian physics, generalized similarities are similarity or skew-similarity relations that subsume EP-inducing unitary and anti-unitary symmetries. The three basic forms are pseudo-Hermiticity,

$g(x)=a e^{-bx}+c$8

pseudo anti-Hermiticity,

$g(x)=a e^{-bx}+c$9

and self skew-similarity,

$s_1,s_2,s_3,s_4$0

These impose spectral maps $s_1,s_2,s_3,s_4$1, $s_1,s_2,s_3,s_4$2, and $s_1,s_2,s_3,s_4$3, respectively, and reduce the codimension of exceptional points from $s_1,s_2,s_3,s_4$4 to $s_1,s_2,s_3,s_4$5 or $s_1,s_2,s_3,s_4$6 depending on the case. The paper’s central claim is that codimension reduction does not fundamentally require unitary or anti-unitary symmetry; the weaker similarity conditions suffice (Montag et al., 2024).

In graph databases, generalized similarity is defined by structural robustness under invertible schema mappings induced by tgds and egds over conjunctive regular path queries. RelSim achieves this by extending ordinary meta-paths to rich relationship expressions

$s_1,s_2,s_3,s_4$7

where $s_1,s_2,s_3,s_4$8 is a nested operator and $s_1,s_2,s_3,s_4$9 is a skip operator. The resulting PathSim-like score is invariant under a broad class of information-preserving structural transformations, so similarity becomes representation-independent rather than tied to a single graph schema (Chodpathumwan et al., 2015).

Coding theory provides a particularly explicit case of generalized similarity between two metric formalisms. For a code $a\approx b$0,

$a\approx b$1

is the rank-metric analogue of Wei’s generalized Hamming weights, and in fact

$a\approx b$2

This allows Hamming-weight bounds and proofs to transfer directly to generalized rank weights, while rank-metric equivalences play the role of monomial Hamming isometries. Operationally, the two theories govern error and erasure correction and information leakage in formally parallel ways, with generalized rank weights doing for linear network coding what generalized Hamming weights do for wiretap channels of type II and code-based secret sharing (Martínez-Peñas, 2015).

6. Similarity operators in modern learning systems

In generalized few-shot semantic segmentation, generalized similarity is implemented as a probabilistic transition from base-label posteriors to novel-label posteriors. If $a\approx b$3 is the frozen base classifier output, then novel prediction is modeled by

$a\approx b$4

with entries

$a\approx b$5

assembled into a column-stochastic similarity transition matrix

$a\approx b$6

To reduce parameters, the matrix is factorized as

$a\approx b$7

and then extended to $a\approx b$8 with base-to-base and base-to-novel blocks so that the transition operator can mitigate catastrophic forgetting while transferring base knowledge to novel classes (Wang et al., 2024).

The same work emphasizes that similarity and imbalance should be decoupled: $a\approx b$9 is intended to capture semantic or visual correspondence between labels, whereas label skew is handled separately by LDAM and transductive inference. This is a notable shift from metric similarity toward operator-valued similarity between label spaces. A plausible implication is that generalized similarity in modern learning is increasingly treated as a learned transport law between representations, posteriors, or classes rather than only as a pairwise score.

Across the cited literature, generalized similarities therefore form a technical vocabulary for controlled relaxation. They may enlarge the admissible symmetries of a Hamiltonian, the allowable overlap maps of a self-similar set, the support objects underlying coding-theoretic weights, the term language used to compare algebraic structures, the path language used in graph databases, or the label-space transitions used in few-shot learning. What remains constant is not a single formula but a structural ambition: to preserve a meaningful notion of sameness while broadening the class of admissible correspondences.