Source-linked AI summary
On Similarity
Luciano da F. Costa
TL;DR
Objective similarity measures are needed to compare mathematical structures, yet familiar approaches share product-based constructions. This paper derives yielding indices from the Kronecker delta, generalizes them across structures, and relates Jaccard forms to the scalar indices.
Problem
Objective comparison supports choices among models and explanations, while familiar similarity measures are product-based despite many possible alternative constructions.
Source: §1 · p. 1Method
The paper uses the Kronecker delta as a scalar reference, derives yielding indices and signed versions, and generalizes them to multisets, vectors, and functions.
Source: §1 · pp. 1–2Results
Takeaways & Limitations
The generalized indices can be employed to define convolution and correlation operations for function comparisons.
Source: §1 · p. 2The work focuses on similarity measurements, leaving extension of its results to distances as immediate rather than developing it.
Source: §1 · p. 1
Abstract
from arXiv · showhide
The objective quantification of similarity between two mathematical structures constitutes a recurrent issue in science and technology. In the present work, we developed a principled approach that took the Kronecker's delta function of two scalar values as the prototypical reference for similarity quantification and then derived for more yielding indices, three of which bound between 0 and 1. Generalizations of these indices to take into account the sign of the scalar values were then presented and developed to multisets, vectors, and functions in real spaces. Several important results have been obtained, including the interpretation of the Jaccard index as a yielding implementation of the Kronecker's delta function. When generalized to real functions, the four described similarity indices become respective functionals, which can then be employed to obtain associated operations of convolution and correlation.
1 Introduction
Objective comparison matters for choosing among models and explanations, but similarity measures span many mathematical structures and possible constructions. The paper develops scalar indices from the Kronecker delta and extends them to multisets, vectors, and real functions.
1 Introduction
- Similarity measures are needed to compare structures used in model validation, but familiar measures such as Euclidean distance, cosine similarity, and inner product are product-based.The paper notes that many other possible measures use minimum, maximum, or absolute values.Source: §1 · p. 1
- The paper takes the Kronecker delta as a scalar-similarity reference and relaxes it into four indices, three normalized to [0, 1].Signed versions add information about the relative alignment of the compared values.Source: §1 · pp. 1–2
- The indices are generalized to multisets, vectors, and real functions, while analogous extensions to matrices, graphs, and scalar or vector fields are not developed.The paper also presents generalized interiority and coincidence indices for functions.Source: §1 · p. 2
- Existing function-similarity work includes interiority and coincidence indices, with coincidence reported as performing well in pattern recognition and as a joint-variation quantifier.The cited examples include pattern recognition and joint variation between random variables.Source: §1 · p. 2
- For real functions, the four indices become functionals that can be combined and used to define convolution and correlation operations.The paper presents these operations as extensions of the function-level similarity framework.Source: §1 · p. 2
- The Jaccard index is linked to the scalar indices s1 and s2 through its multiset forms, including coincidence and addition-based Jaccard indices.The paper identifies these as generalizations of the scalar indices.Source: §1 · p. 2
2 Pairwise Similarities in R
The section starts from identity as the strictest similarity criterion, then relaxes it to yield more tolerant indices for real values. It also extends the indices to account for sign and compares their behavior.
2 Pairwise Similarities in R
- The strict identity-based similarity approach assigns 1 when x = y and 0 otherwise; because it is too strict, tolerance must be implemented in quantification.Source: §2 · p. 2
- The case x = 0 and y = 0 is excluded, so applications where both values can be zero require additional checking and handling.Source: §2 · p. 2
- The indices lose information about relative signs; signed generalizations use whether x and y point in the same or opposite directions.The signed indices s1, s2, and s3 lie in [-1, 1], while s4 lies in [-∞, ∞].Source: §2 · pp. 3–4
- s1 and s2 are more strict, while s3 and s4 are more yielding blurred versions of the Kronecker-delta similarity.In Figure 4, s1 and s2 peak along the main diagonal; the peak is less marked for s3 and virtually indistinguishable for s4.Source: §2 · p. 4, Figure 4 · p. 4
- s4 does not penalize values near y = −x, a property linked to Pearson correlation’s tendency to overestimate joint variation when variables are not strongly related.The passage also links that tendency to.Source: §2 · p. 4
- s1 and s2 are closely and bijectively associated, whereas s3 and s4 are not bijectively associated with each other or with the first two indices.Figure 5 presents scatterplots for the pairwise associations.Source: §2 · p. 4, Figure 5 · p. 4
3 Multiset Similarities
The scalar similarity indices are extended to multisets, with corresponding definitions for union and intersection. The resulting generalized Jaccard index connects the multiset construction to the Kronecker-delta-based scalar similarity.
3 Multiset Similarities
- The four scalar similarity indices can be applied to multisets, using elementwise multiplicities to quantify similarity.For multisets sharing support, union uses the maximum multiplicity and intersection the minimum; differing supports can be combined by taking the union of their supports.Source: §3 · p. 5
- For multisets, s1 generalizes the Jaccard similarity index to negative values [1, 2], while s2 is identical to the recently proposed addition-based multiset Jaccard index [1, 2].Source: §3 · p. 5
- The generalized Jaccard index for multisets is a smoothed implementation of Kronecker-delta-based similarity, extending that interpretation to scalars, multisets, vectors, and scalar fields.The multiset developments also derive recently introduced generalized Jaccard indices.Source: §3 · p. 5
4 Vector Similarities
The multiset similarity indices extend directly to vectors because vectors can be represented as multisets on a shared index support. In this setting, s4 is the inner product.
5 Function Similarities
The multiset-based indices extend to real functions, becoming functionals. These functionals support associated elementwise operations, and the sine–cosine example illustrates how the resulting quantities represent signed similarity.
5 Function Similarities
- The four similarity indices generalize to real functions through the continuous representation of multisets.The resulting indices are functionals and can be combined to implement convolution and correlation operations between functions.Source: §5 · pp. 5–6
- For real functions, the functionals can be interpreted as binary operations between functionals, with related elementwise operations.The s1 functional uses separate integrals in its numerator and denominator.Source: §5 · p. 6
- For sine and cosine over one complete period, the common-area operation gives their common area, the maximum operation gives the maximum of their absolute values, and s1 indicates signed similarity.Source: §5 · p. 6, Figure 6 · p. 6
6 The Interiority and Coincidence Indices
This section addresses how the Jaccard index can be supplemented to capture how much one set is interior to another. It presents generalized interiority and coincidence indices for multisets, vectors, and functions, including a same-sign restriction for multiset interiority.
6 The Interiority and Coincidence Indices
- The coincidence index supplements the traditional Jaccard index, which does not capture how much one set is interior to the other, by multiplying Jaccard by interiority.The work also considers generalized versions of both indices.Source: §6 · pp. 6–7
- Interiority and coincidence are treated as similarity indices and considered for generalization to multisets, vectors, and functions.Source: §6 · p. 7
- For multiset interiority, S+ includes only pairs of same-sign multiplicities, giving 0 ≤ I(A, B) ≤ 1; whole support may be used when needed, such as for template matching.Source: §6 · p. 7
7 Similarity Convolutions and Correlations
The section treats the generalized similarity indices as functionals on real function spaces and uses them to define associated convolutions and correlations. It also connects these operations to standard convolution, Walsh functions, and template-matching and filtering tasks.
7 Similarity Convolutions and Correlations
- Each similarity index generalized to real function spaces is a functional that can be used to obtain associated convolutions and correlations.Source: §7 · p. 7
- The convolution associated with s4 is a normalized version of the standard convolution.Source: §7 · p. 7
- The common product is associated with Walsh functions, enabling analogous fast computing schemes for convolutions.The passage compares these schemes with the fast Fourier transform.Source: §7 · p. 7
- In operations related to the common product, maximum is the nonlinear counterpart of the Fourier transform's bilinear products; related correlations show substantially enhanced performance in template matching and filtering.Source: §7 · p. 7
8 Concluding Remarks
The paper develops similarity indices from the Kronecker delta and extends signed versions to multisets, vectors, and functions. It relates these indices to Jaccard and to standard inner-product operations, while framing similarity quantification as important across applied fields.
8 Concluding Remarks
- Quantifying similarity between mathematical entities is described as ubiquitous in applied areas including pattern recognition, signal processing, and machine intelligence.Source: §8 · p. 7
- The paper takes the Kronecker delta as a reference for scalar similarity and develops four more yielding indices, three bounded by [0, 1], plus signed versions bounded by [−1, 1].Source: §8 · p. 8
- The four signed indices are extended to multisets, vectors, and functions, with analogous extensions also possible for scalar and vector fields.Source: §8 · p. 8
- The Jaccard index relates directly to s1, while s4 leads to the standard inner-product functional and its convolution and correlation.The addition-based multiset Jaccard index follows from s2.Source: §8 · p. 8