Title: Two applications of Min-Max-Jump distance
Venue: NeurIPS 2024
Accepted: no
Keywords: distance

Silhouette coefficient

Davies–Bouldin index

Calinski-Harabasz index

K-means

clustering

clustering evaluation

minimax path problem
Abstract: We explore two applications of Min-Max-Jump distance (MMJ distance): MMJ-based K-means and MMJ-based internal clustering evaluation index.  K-means and its variants are possibly the most popular clustering approach. A key drawback of K-means is that it cannot deal with data sets that are not the union of well-separated, spherical clusters. MMJ-based K-means proposed in this paper overcomes this demerit of K-means, so that it can handle irregularly shaped clusters. Evaluation (or "validation") of clustering results is fundamental to clustering and thus to machine learning. Popular internal clustering evaluation indices like Silhouette coefficient, Davies–Bouldin index, and Calinski-Harabasz index performs poorly in evaluating irregularly shaped clusters. MMJ-based internal clustering evaluation index uses MMJ distance and Semantic Center of Mass (SCOM) to revise the indices, so that it can evaluate irregularly shaped data. An experiment shows introducing MMJ distance to internal clustering evaluation index, can systematically improve the performance. We also devise two algorithms for calculating MMJ distance.

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Abstract

We explore two applications of Min-Max-Jump distance (MMJ distance): MMJbased K-means and MMJ-based internal clustering evaluation index. K-means and its variants are possibly the most popular clustering approach. A key drawback of K-means is that it cannot deal with data sets that are not the union of well-separated, spherical clusters. MMJ-based K-means proposed in this paper overcomes this demerit of K-means, so that it can handle irregularly shaped clusters. Evaluation (or "validation") of clustering results is fundamental to clustering and thus to machine learning. Popular internal clustering evaluation indices like Silhouette coefficient, Davies-Bouldin index, and Calinski-Harabasz index performs poorly in evaluating irregularly shaped clusters. MMJ-based internal clustering evaluation index uses MMJ distance and Semantic Center of Mass (SCOM) to revise the indices, so that it can evaluate irregularly shaped data. An experiment shows introducing MMJ distance to internal clustering evaluation index, can systematically improve the performance. We also devise two algorithms for calculating MMJ distance.

1 Introduction

Distance is a numerical measurement of how far apart objects or points are. It is usually formalized 16 in mathematics using the notion of a metric space. A metric space is a set together with a notion of 17 distance between its elements, usually called points. The distance is measured by a function called 18 a metric or distance function. Metric spaces are the most general setting for studying many of the 19 concepts of mathematical analysis and geometry. 20

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In this paper, we introduce two algorithms for calculating Min-Max-Jump distance (MMJ distance) and explore two applications of it. Including MMJ-based K-means (MMJ-K-means) and MMJ-based internal clustering evaluation index.

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MMJ-K-means improves K-means, so that it can handle irregularly shaped clusters. We claim MMJCH is the SOTA (state-of-the-art) internal clustering evaluation index, which achieves an accuracy of 90 / 145 . MMJ-CH is one of the MMJ-based internal clustering evaluation indices.

2 RELATED WORK

2.1 Different distance metrics

Many distance measures have been proposed in literature, such as Euclidean distance or cosine 29 similarity. These distance measures often be found in algorithms like k-NN, UMAP, HDBSCAN, 30 etc. The most common metric is Euclidean distance. Cosine similarity is often used as a way to 31 counteract Euclidean distance's problem in high dimensionality. The cosine similarity is the cosine 32 of the angle between two vectors. 33

Hamming distance is the number of values that are different between two vectors. It is typically used 34 to compare two binary strings of equal length (1). 35

Manhattan distance is a geometry whose usual distance function or metric of Euclidean geometry 36 is replaced by a new metric in which the distance between two points is the sum of the absolute 37 differences of their Cartesian coordinates (2). 38

Haversine distance is the distance between two points on a sphere given their longitudes and latitudes. 45 It is similar to Euclidean distance in that it calculates the shortest path between two points. The main 46 difference is that there is no straight line, since the assumption is that the two points are on a sphere 47 (6). 48

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2.2 K-means

K-means (7) and its variants (8; 9; 10) are possibly the most well-liked clustering approach. K-means divides the data into K groups, where K is a hyper-parameter to be optimized. It aims to reduce the within-cluster dissimilarity. While popular, K-means and its variants perform poorly for data sets that are not the union of well-separated, spherical clusters. MMJ-based K-means (MMJ-K-means) proposed in this paper overcomes this demerit of K-means, so that it can handle irregularly shaped clusters.

2.3 Internal clustering evaluation index

Evaluation (or "validation") of clustering results is as difficult as the clustering itself (11). Popular approaches involve "internal" evaluation and "external" evaluation. In internal evaluation, a clustering result is evaluated based on the data that was clustered itself. Popular internal evaluation indices are Davies-Bouldin index (12), Silhouette coefficient (13), Dunn index (14), and Calinski-Harabasz index (15) etc. In external evaluation, the clustering result is compared to an existing "ground truth" classification, such as the Rand index (16). However, knowledge of the ground truth classes is almost never available in practice.

In Section 5.2, an experiment shows introducing Min-Max-Jump (MMJ) distance to internal clustering evaluation index, can systematically improve the performance.

2.4 Path-based distances

Euclidean distances are frequently used in machine learning and clustering methods to compare 67 points. However, the distance is data-independent, and not tailored to the geometry of the data. Many 68 metrics that are data-dependent have been devised, such as diffusion distances (17) and path-based 69 distances (18; 19). MMJ distance is a path-based distance. 70

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3 Definition of Min-Max-Jump

Definition 1. Min-Max-Jump distance (MMJ distance)

Ω is a set of points (at least one). For any pair of points p, q ∈ Ω , the distance between p and q is 73 defined by a distance function d(p,q) (such as Euclidean distance). i, j ∈ Ω , Ψ ( i,j,n, Ω) is a path from 74 point i to point j, which has length of n points (see Table 1). Θ ( i,j, Ω) is the set of all paths from point 75

Where ϵ is a path from point i to point j, max _ jump ( ϵ ) is the maximum jump in path ϵ . Π is the set 79 of all maximum jumps. min (Π) is the minimum of Set Π . 80

Set Ω is called the Context of the Min-Max-Jump distance. It is easy to check MMJ ( i, i | Ω) = 0 . 81

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There is a minor difference between Min-Max-Jump distance and other similar distances: Min-Max89 Jump distance stresses the context of the distance. The context is like the condition in conditional 90 probability. The difference becomes non-trivial when we need to calculate the pairwise MMJ distance 91 matrix of a set S , under the context of its superset X , such as in Section 6.3 of (24). A set Ω is a 92 superset of another set B if all elements of the set B are elements of the set Ω . 93

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3.1 An example

Suppose Set Ω is composed of the four points in Figure 1. There are five (non-looped) paths from point a to point c in Figure 1:

According to Definition 1, MMJ ( a, c | Ω) = 12 .

To understand Min-Max-Jump distance, imagine someone is traveling by jumping in Ω . Suppose MMJ ( i, j | Ω) = δ . If the person wants to reach j from i , she must have the ability of jumping at least δ . Otherwise, j is unreachable from i for her. Whether the distance to a point is "far" or "near" is measured by how far (or how high) it requires a person to jump. If the requirement is large, then the point is "far", otherwise, it is "near."

3.2 Properties of MMJ distance

Theorem 1. Suppose i, j, p, q ∈ Ω , then,

where d(x,y) is a distance function (Table 1).

Proof. MMJ ( i, j | Ω) = δ is equivalent to ∃ P ∈ Θ ( i,j, Ω) , such that M ( P ) = δ , and ∀ T ∈ Θ ( i,j, Ω) , M ( T ) ≥ δ , where Θ ( i,j, Ω) is the set of all paths from point i to point j under context Ω . M ( P ) is the maximum jump in path P . We can assume MMJ ( p, q | Ω) > δ and MMJ ( p, q | Ω) < δ , then we will arrive to a contradiction in both cases.

Theorem 2. Suppose r ∈ { 1 , 2 , . . . , n } , 118

For the meaning of Ω t , Ω r , Ω [1 ,n ] , and Ω [1 ,n +1] , see Table 1.

Proof. There are n possibilities of the MMJ path from Ω n +1 to Ω r , under the context of Ω [1 ,n +1] , set X enumerate them all. Each element of X is the maximum jump of each possibility. Therefore, according to the definition of MMJ distance, MMJ (Ω n +1 , Ω r | Ω [1 ,n +1] ) = min ( X ) .

Corollary 1. Suppose r ∈ { 1 , 2 , . . . , N } , p / ∈ Ω ,

For the meaning of Ω+ p , see Table 1.

Proof. The proof follows the conclusion of Theorem 2.

Theorem 3. Suppose i, j ∈ { 1 , 2 , . . . , n } ,

Proof. There are two possibilities of the MMJ path from Ω i to Ω j , under the context of Ω [1 ,n +1] : Ω n +1 is in the path or it is not in the path. x 2 is the min-max jump of the first possibility; x 1 is the min-max jump of the second possibility. Therefore, according to the definition of MMJ distance, MMJ (Ω i , Ω j | Ω [1 ,n +1] ) = min ( x 1 , x 2 ) .

4 Calculation of Min-Max-Jump distance

We propose two methods to calculate the pairwise Min-Max-Jump distance matrix of a dataset. There are other methods for calculating or estimating it, such as a modified SLINK algorithm (25), or with Cartesian trees (26; 27), or from a sequence of nearest neighbor graphs (23), or a modified version of the Floyd-Warshall algorithm.

4.1 MMJdistance by recursion

The first method calculates M Ω by recursion. M Ω is the pairwise MMJ distance matrix of Ω (Table 1). M k, Ω [1 ,k ] is the MMJ distance matrix of the first k points of Ω (Table 1). Note M 2 , Ω [1 , 2] is simple to calculate. M Ω = M N, Ω [1 ,N ] . M Ω is a N × N symmetric matrix. Rows and columns of M Ω are indexed from 1 to N.

Step 7 of Algorithm 1 can be calculated with the conclusion of Theorem 2; Step 12 of Algorithm 1 can be calculated with the conclusion of Theorem 3.

Algorithm 1 has complexity of O ( n 3 ) , where n is the cardinality of Set Ω .

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Algorithm 1 MMJ distance by recursion

4.2 MMJdistance by calculation and copy

According to the conclusion of Theorem 1, there are many duplicated values in M Ω . So in the second method we can calculate the MMJ distance value in one position and copy it to other positions in M Ω .

A well-known fact about MMJ distance is: "the path between any two nodes in a minimum spanning tree (MST) is a minimax path." A minimax path in an undirected graph is a path between two vertices v , w that minimizes the maximum weight of the edges on the path. That is to say, it is a MMJ path. By utilizing this fact, we propose Algorithm 2.

Algorithm 2 MMJ distance by Calculation and Copy

Input: Ω Output: M Ω 1: function MMJ_CALCULATION_AND_COPY( Ω ) 2: Initialize M Ω with zeros 3: Construct a MST of Ω , noted T 4: Sort edges of T from large to small, generate a list, noted L 5: for e in L do 6: Remove e from T . It will result in two connected sub-trees, T 1 and T 2 ; 7: Traverse T 1 and T 2 ; 8: For all pair of nodes ( p, q ) , where p ∈ T 1 , q ∈ T 2 . Fill in M Ω [ p, q ] and M Ω [ q, p ] with the weight of e . 9: end for 10: return M Ω 11: end function

The complexity of Algorithm 2 is O ( n 2 ) . Because the construction of a MST of a complete graph is 160 O ( n 2 ) . During the "for" part (Step 5 to 9) of the algorithm, it accesses each cell of M Ω only once. 161 Unlike Algorithm 1, which accesses each cell of M Ω for O ( n ) times. The merit of the "Calculation 162 and Copy" method is that it is easier to understand than using the Cartesian trees (26; 27). 163

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(d) data A, Standard K-means

(e) data B, Standard K-means

(f) data C, Standard K-means

(g) data A, MMJ-K-means

(h) data B, MMJ-K-means

(i) data C, MMJ-K-means

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5 Applications of Min-Max-Jump distance

We explore two applications of MMJ distance, and test the applications with experiments. All the MMJ distances in the experiments are calculated with Algorithm 1.

5.1 MMJ-based K-means

K-means clustering aims to partition n observations into k clusters in which each observation belongs 168 to the cluster with the nearest mean (cluster center or centroid), serving as a prototype of the cluster 169 (28). Standard K-means uses Euclidean distance. We can revise K-means to use Min-Max-Jump 170 distance, with the cluster centroid replaced by the Semantic Center of Mass (SCOM) (particularly, 171 One-SCOM) of each cluster. For the definition of SCOM, see a previous paper (29). One-SCOM is 172 like medoid, but has some difference from medoid. Section 6.3 of (29) compares One-SCOM and 173 medoid. In simple terms, the One-SCOM of a set of points, is the point which has the smallest sum 174 of squared distances to all points in the set. 175

Standard K-means usually cannot deal with non-spherical shaped data, such as the ones in Figure 2. 176

MMJ-based K-means (MMJ-K-means) can cluster such irregularly shaped data. Figure 2 compares 177

Standard K-means and MMJ-K-means, on clustering three data which come from the scikit-learn 178 project (30). Figure 3 are eight more samples of MMJ-K-means. The data sources corresponding to 179 the data IDs can be found at this URL (temporarily hidden for double blind review). 180

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It can be seen MMJ-K-means can (almost) work properly for clustering the 11 data, which have different kinds of shapes. The black circles are Border points (Definition 2), the red stars are the center (One-SCOM) of each cluster. During training of MMJ-K-means, the Border points are randomly allocated to one of its nearest centers.

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Definition 2. Border point

A point is defined to be a Border point if its nearest mean (center, centroid, or One-SCOM) is not unique.

Compared with other clustering models that can handle irregularly shaped data, such as Spectral 188 clustering or the Density-Based Spatial Clustering of Applications with Noise (DBSCAN), the merit 189 of MMJ-K-means is its simplicity; the logic of MMJ-K-means is as simple as K-means. We just 190 replace the Euclidean distance with MMJ distance, and the centroid with the Semantic Center of 191 Mass (SCOM). 192

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5.2 MMJ-based internal clustering evaluation index 193

Calinski-Harabasz index, Silhouette coefficient, and Davies-Bouldin index are three of the most 194 popular techniques for internal clustering evaluation. They are used to calculate the goodness of a 195 clustering technique. 196

The Silhouette coefficient for a single sample is given as: 197

where a is the mean distance between a sample and all other points in the same class. b is the mean 198 distance between a sample and all other points in the next nearest cluster. The Silhouette coefficient 199 for a set of samples is given as the mean of Silhouette coefficient for each sample. 200

We can also revise Silhouette coefficient to use Min-Max-Jump distance, forming a new internal 201 clustering evaluation index called MMJ-based Silhouette coefficient (MMJ-SC). We tested the 202 performance of MMJ-SC with the 145 datasets mentioned in another paper(31). MMJ-SC obtained 203 a good performance score compared with the other seven internal clustering evaluation indices 204 mentioned in the paper(31). Readers can check Table 2 and compare with Table 5 of Liu's paper(31). 205

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MMJ-based Calinski-Harabasz index (MMJ-CH) and MMJ-based Davies-Bouldin index (MMJ-DB) were also tested. In calculation of these two indices, besides using MMJ distance, the center/centroid of a cluster is replaced by the One-SCOM of the cluster again, as in MMJ-K-means. It can be seen that MMJ distance systematically improves the three internal clustering evaluation indices (Table 2). The best performer is MMJ-CH, which achieves an accuracy of 90 / 145 . The accuracy of an index is computed by evaluating the index's ability of recognizing the best partition of a dataset from hundreds of candidate partitions(31).

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5.2.1 Using MMJ-SC in CNNI

The Clustering with Neural Network and Index (CNNI) model uses a Neural Network to cluster 214 data points. Training of the Neural Network mimics supervised learning, with an internal clustering 215 evaluation index acting as the loss function (24). CNNI with standard Silhouette coefficient as the 216 internal clustering evaluation index, cannot deal with non-flat geometry data, such as data B and 217 data C in Figure 2. MMJ-SC gives CNNI model the capability of processing non-flat geometry data. 218 E.g., Figure 4 is the clustering result and decision boundary of data B by CNNI using MMJ-SC. 219 It uses Neural Network C of the CNNI paper (24). CNNI equipped with MMJ-SC, achieves the 220

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first inductive clustering model that can deal with non-flat geometry data (24). For the definition of 221 non-flat geometry data, see this 1 Stackexchange question. 222

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6 Discussion

6.1 Using PAM

Since One-SCOM is like medoid, in MMJ-K-means, we can also use the Partitioning Around Medoids (PAM) algorithm or its variants to find the One-SCOMs (32).

6.2 Multiple One-SCOMs in one cluster

There might be multiple One-SCOM points in a cluster, which have the same smallest sum of squared distances to all the points in the cluster. Usually they are not far from each other. We can arbitrarily choose one or keep them all. If we keep them all, then the One-SCOM of a cluster is not a point, but a set of points. If the One-SCOM is a set, when calculating a point's MMJ distance to the One-SCOM of a cluster, we can select the minimum of the point's MMJ distances to all the One-SCOM points.

6.3 Differentiating border points

Border points defined in Definition 2 can further be differentiated as weak and strong border points. Definition 3. Weak Border Point (WBP)

A point is defined to be a WBP if its nearest mean (center or One-SCOM) is not unique but less than K , where K is the number of clusters.

Definition 4. Strong Border Point (SBP)

A point is defined to be a SBP if its nearest mean (center or One-SCOM) is not unique and equals K , where K is the number of clusters.

Then we can process different kinds of border points with different strategies. E.g., deeming the Strong Border Points as outliers and removing them.

7 Conclusion and Future Works

We proposed two algorithms for calculating Min-Max-Jump distance (MMJ distance), and tested 244 two applications of it: MMJ-based K-means and MMJ-based internal clustering evaluation index. 245 MMJ-K-means overcomes a big drawback of K-means, improving its ability of clustering, so that it 246 can handle irregularly shaped clusters. We claim MMJ-CH is the SOTA (state-of-the-art) internal 247 clustering evaluation index, which achieves an accuracy of 90 / 145 . To thoroughly test the internal 248 clustering evaluation indices, we conducted an experiment on a set of 145 datasets. A normal 249 Machine Learning paper usually uses several or dozens of datasets to test their models or algorithms. 250 In summary, MMJ distance has good capability and potentiality in Machine Learning. Further 251 research may test its applications in other models, such as other clustering evaluation indices. 252

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NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

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Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [No]

Justification: We have used 145 datasets to test the models in the paper. Maybe it is not enough, we need more datasets to test the models.

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Peer reviews:
comments: SUMMARY:
The paper proposes to use a new distance, Min-Max-Jump, which is the minimum largest distance on any path between two points, to be used in k-means clustering to learn clusters.

STRENGTHS:
The distance can overcomes some demerits of the convex ("spherical" in the paper) clusters.

WEAKNESSES:
The distance in the paper is, in fact, related to single linkage clustering that assign give a pair of points a distance at which the pair is joint to one cluster. This need to be analyzed to relate to previous work as well as to compute pairwise distances efficiently. 

Theoretical property of the distance is poor. The paper should review many other density-based distance functions to put this work into the correct context. 

There would be a lot of problems using this distance as many of pairs of nodes would share the same distance. There is no analysis on the  metric property of this distance. 

On evaluation, the methods need to compare to single-linkage clustering (SSL) at the very least as all the advantages of using this distance with k-means are available in SLL in its simplest form.

Presentation-wise, it is hardly up the standard. There are methods/algorithms/concepts that are mentioned as "a is like b with a difference" without a formal definition. This mixes up definitions and properties.

QUESTIONS:
N/A

LIMITATIONS:
The paper uses a new distance without theoretical justifications. It learns nonconvex clusters by using a nonconvex clustering-based distance (without explicitly mentioning it), which is hardly a novelty.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2024-07-09

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new metric, min-max jump distance. Effectively, say we are given a complete graph with vertex set $\\Omega$ and edge weights $d(x,y)$ denoting the distance between $x$ and $y$, where $d$ is a metric. Then $MMJ(x,y|\\Omega')$ is the minimum, over all paths between $x$ and $y$, of the maximum weight edge between $x$ and $y$ on the subgraph induced on the vertices $\\Omega' \\subseteq \\Omega$. Explained nicely in the paper, if you started at vertex $x$ and wanted to get to $y$ and $d(\\cdot,\\cdot)$ denoted the distance required to ``jump'' from one vertex to another, what is the minimum distance you need to be able to jump to somehow traverse from $x$ to $y$?\n\nThis is a nice intuitive metric, and has strong connections to the minimum spanning tree. In fact, I suspect there is more literature to draw from minimum spanning tree research that could yield conclusions about MMJ. The MST is also nice in clustering since oddly shaped clusters (non-convex, for instance) can have small MSTs. This is the idea of MMJ: use it as a metric for K-means so that it can identify non-spherical clusters.\n\nThe paper proves some notable theory about the properties of MMJ. Mostly, they show how: 1) When adding a new vertex $p$ to a set $\\Omega$, $p$'s MMJ within the context of $\\Omega+p$ can be computed knowing all pairwise MMJ's within $\\Omega$ within the context of $\\Omega$. This effectively adds a new point and evaluates it within the complete, updated context. 2) Given the MMJs for this additional point $p$ within the context of $\\Omega +p$, expand the MMJ context of all other pairs in $\\Omega$ to the context of $\\Omega+p$.\n\nThis can then be used in a very dynamic programming-like manner to start with just two points, add new points $p$ and find the context of $p$ and all other points, and then update all known existing MMJs to the new context. This is their algorithm 1, requiring $O(n^3)$ time. They also use properties of the MST to bypass unnecessary calculation to yield algorithm 2, which takes only $O(n^2)$ time (to find the MST).\n\nThey then evaluate the performance of algorithms using the MMJ measure on irregular shaped clusters to verify that MMJ helps identify these. This makes sense, since they likely have small MSTs, but not small average/sum/max/etc distances within clusters. Notably, they show how MMJ improves K-means.\n\nSTRENGTHS:\nI think MMJ is a very cool metric with nice properties and intuition, particularly that related to the MST (I wish the authors had spent more time discussing this!). Their findings are nice and relatively simple to understand (in spite of the presentation). They show that it helps K-means expand to more complicated cluster shapes, and overall it is a very nice, NeurIPS-worthy result. However, as I will explain in the weaknesses section, I do not think this paper is in an acceptable state for NeurIPS.\n\nWEAKNESSES:\nThere are a few notable downsides. In terms of the result, I'm not entirely convinced of its novelty. How much of this is actually a re-iteration of MST-based methods already understood? Is this really better than other MST-based algorithms on irregular clusters (think single linkage)? I know that there is a lot of literature that explores irregular shaped clusters, but I am not an expert in this area and so I cannot place this work in the context of existing results. I wish the authors would explain that. Though even if these algorithms aren't entirely better than state of the art, the novelty of the nice formulation of MMJ is certainly appreciated.\n\nHowever, the biggest flaw in the paper is the writing quality. There are places where the paper is nice and concise, but most of the time it just lacks exposition to understand the higher level of things or adequate details to fully understand what is happening. Formal proofs are contained in the paper, but the jumps in some of the proofs are too large. Theorems and proofs are placed back to back with no high level explanation. Algorithms are written and pseudocode with only the briefest justifications, and no thorough explanations. This is not an acceptable paper for NeurIPS, and I think these issues are too extensive to simply ask the authors to revise. Though if other commenters disagree, I am amenable to changing my opinion.\n\nAnd one of the disappointing things about this is how natural this work is and how much it lends itself to nice intuitive explanations and visual depictions! For instance, you could do some very nice visualizations of Algorithm 1, where you depict a matrix and show which indices have been calculated to what context $\\Omega_n$ at each time point. This clarifies the different purpose of the two loops.\n\nI hope to see this paper submitted again later in a more cleaned up state!\n\nQUESTIONS:\n1. What exactly is a path-based distance?\n\n2. When you define MMJ, you do not specify exactly what a path is. All you introduce is a point set and a distance function. Are we to assume that a \"path\" is done on the complete graph with edge weights equal to the distances between points? Please clarify in the paper. I read the paper with this assumption.\n\n3. How would MMJ compare to the similarly defined max-min jump measure? I would think it would be more akin to a similarity measure than a distance.\n\n4. Line 114: what is M(P)? Also, how do you arrive at the contradiction in this proof? This needs more explanation!\n\n5. Line 122: What do you mean there are n possibilities? Aren't there something like n! candidate paths, since you check all paths of all lengths with a fixed set of endpoints?\n\n6. Thm 2 proof: You say X enumerates them all - what about the direct path that just traverses the single edge between the points?\n\n7. Is Corollary 1 actually any different from Thm 2? It seems like in both you just have a set $\\Omega_{[1,n]}$ and then you have a point outside of the set that you want to get the MMJ to (with respect to some reference point). I think Corollary 1 is just a cleaner way to state Theorem 2.\n\n8. In Theorem 3, is it actually true that $x_2$ is the MMJ if $\\Omega_{n+1}$ is in the path? It doesn't break your argument, but if the solution path doesn't go through $\\Omega_{n+1}$, then it's possible that the MMJ path from i to n+1 and from j to n+1 share an edge. So the MMJ path from i to j that goes through n+1 could actually be worse since we aren't allowed to traverse that edge twice. Again, the proof still works, but I think that claim only holds if the MMJ path from i to j does, in fact, go through n+1. Otherwise, we might have something larger, which is ignored by taking the minimum.\n\n9. Section 4.1: Why does k appear twice in the matrix subscripts? Couldn't it just be something like $M_k$?\n\n10. How does MMJ-K-means perform on spherical data compared to K-means? Presumably, worse. It would be nice to quantify how much worse.\n\nLIMITATIONS:\nNone notable", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents the Min-Max-Jump (MMJ) distance concept and two calculation methods, focusing on path optimization in data analysis and clustering. The contributions include introducing MMJ distance, proposing efficient calculation methods, discussing its properties and applications, and offering a user-friendly approach for practical implementation. Overall, the paper introduces a new distance metric for path optimization and data analysis, providing useful tools and insights for various applications in the field.\n\nSTRENGTHS:\nS1: The paper demonstrates strength through its meticulous use of theorems and proofs, enhancing the credibility and robustness of the research findings.\n\nS2: Clear visualization of results in the paper aids in effectively conveying complex information to the readers, improving understanding and interpretation.\n\nS3: Extensive literature citations throughout the paper showcase a strong foundation of existing knowledge and research, adding depth and scholarly rigor to the study's presentation.\n\nWEAKNESSES:\nW1: The paper's writing style deviates from academic norms, indicating a need for improvement in writing proficiency.\n\nW2: The extremely brief Introduction lacks a detailed definition of the problem, its significance, and challenges. Moreover, it lacks citation support for the points presented. While Section 2.1 mentions methods like k-NN, UMAP, HDBSCAN, it fails to provide corresponding references, lacking essential academic backing.\n\nW3: The overall structure of the paper lacks clarity, as it introduces different distance metrics in Section 2.1 but introduces a new distance measurement approach in Section 2.4, leading to disjointed logic.\n\nW4: The presentation of various distance metrics in Section 2.1 appears disorganized and lacks coherence.\n\nW5: The extensive definition provided towards the end of Section 6.3 disrupts the logical flow of the paper, suggesting a need to adjust the paper's structural coherence.\n\nQUESTIONS:\nQ1:Can the authors provide more insights into the practical implications of their research findings and how they can be applied in real-world scenarios?\n\nLIMITATIONS:\nThe paper does not discuss limitations. The authors seem to perceive their work as solely testing models with datasets without considering the shortcomings of the algorithms themselves.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

comments: SUMMARY:
Different distance metrics have been introduced in the literature for data analysis. In this paper the authors consider the min-max-jump distance and apply it in the context two applications, namely, k-means clustering and as an internal clustering evaluation index. They also present two algorithms for computing the min-max-jump distance.

Experimental comparisons reveal that min-max-jump based k-means clustering is better than standard k-means clustering. Also, the min-max-jump distance is shown to be a better internal clustering evaluation index.

This referee feels that this work is rather incremental. Also, experiments have been conducted only on very small datasets.

STRENGTHS:
The authors demonstrate the efficacy of the min-max-jump distance on two different applications.

Experimental results have also been supplied to support their assertions.

WEAKNESSES:
The work done is incremental with very limited novelty.
Extensive experiments are called for.

QUESTIONS:
None

LIMITATIONS:
None
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12