# Questions tagged [similarity]

The similarity tag has no usage guidance.

33
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### How can I measure similarity between two graphs with identical topology but different edge weights

I have two graphs, G1 and G2, with exactly the same topology. Their only difference lies in the edge weights, which vary between 0 and 1.
How can I measure the similarity between G1 and G2 under these ...

6
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0
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230
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### Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?

QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...

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109
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### What is the status of The Halmos Similarity Problem?

What is the general status of "The Halmos Similarity Problem"(HSP) in Operator theory?For What conditions ,HSP has been solved?

8
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262
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### How to check two matrices for similitude over $\mathbb{Z}$?

General question. Let $A$ and $B$ be two $n\times n$-matrices over
$\mathbb{Z}$. How do I algorithmically check whether $A$ and $B$ are similar
(i.e., conjugate in the ring $\mathbb{Z}^{n\times n}$)?
...

3
votes

1
answer

215
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### Comparing two distributions based of the ratio of their moments

I am looking for some metric for distribution with support on the interval $[0,1-\epsilon]$, that will be based on the ratio of their moments.
That is, if $X\sim f(x)$, $Y\sim g(y)$, I'm looking for a ...

1
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0
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178
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### What is a good algorithm to measure similarity between isomorphic graphs with different node labels?

I am using graphs to represent some structured data. In my case, I have a time series of undirected unweighted graphs with the same topology (i.e. isomorphic graphs with same number of nodes and edges,...

0
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72
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### Fast decay of eigenvector elements

Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...

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57
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### What is the name given to the solution to the equation $cU = Y U Y$ for a given symmetric, positive definite, real-valued matrix $Y$

Overarching question is: What is the name given to the solution to the equation $cU = Y U Y$ for a given symmetric, positive definite, real-valued matrix $Y$? And what procedure is used to solve this ...

4
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218
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### Studying finite groups with Euclidean geometry?

Since each finite group $G$ can be considered as a subgroup of the symmetric group, by Cayley's theorem, we might see the elements of $G$ as permutations $\pi$.
Consider for each $\pi \in G$ the set:
...

7
votes

1
answer

495
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### Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?

Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+...

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384
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### Comparison of two similarity matrices

English is not my first language, so please excuse any mistakes.
I'm working with two similarity matrices on the same data set: Suppose I have $n$ items, and I calculated the similarity of each item ...

3
votes

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answers

111
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### Symmetrization of pentadiagonal matrices

Nonsymmetric tridiagonal matrices $T_3$ can easily be symmetrized via a (diagonal) similarity transformation $D=\text{diag}(d_1, \dots, d_n)$ (i.e. see Wikipedia)
$$
J_3=D^{-1} T_3 D \,.
$$
Is there ...

3
votes

1
answer

142
views

### Efficient eigen-decomposition of a real matrix with all real eigenvalues

I'm optimising a radar algorithm that results in real matrices which are not symmetric but which are guaranteed to have real eigenvalues. Each matrix is therefore similar to a symmetric matrix. I am ...

3
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117
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### Sparsest similar matrix

Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...

5
votes

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answers

362
views

### Shapes defined by points

Can shapes determined by some number of points?
From an amazing theorem in plane curves geometry we know that vertices of triangles similar to arbitrary triangle $T$ is dense on every closed jordan ...

1
vote

1
answer

191
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### Are there two tetrahedrons with the same volume that share their opposite edge lengths and arent the same or a different chirality of the same? [closed]

I have been coming up with an efficient way to decide if two tetrahedrons are similar. I believe that it is enough for a computer to check for the ordered by length list of pairs of opposite edges on ...

3
votes

1
answer

168
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### Similarity via symmetric matrix

Let $K$ be a field extension of $F$. If two $n\times n$ matrices $A,B \in M_n(F)$ are similar via a matrix $P \in GL_n(K)$ (that is, $A=PBP^{-1}$), then there exists a matrix $Q\in GL_n(F)$ such that $...

0
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2
answers

146
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### Simultaneous special orthogonal similarity problem

Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that
$$A=...

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63
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### Eigenvalues of sum of two fuchsian matrices

Dear mathoverflow users,
I am trying to solve a problem concerning eigenvalues and sum of matrices.
In particular: consider the expression
$$
A=\frac{E}{x-x_1}+\frac{F}{x-x_2},
$$
and suppose to know ...

0
votes

1
answer

142
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### Sizes and shapes of Dedekind cuts

My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra.
If we define a ...

2
votes

0
answers

447
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### What does similar eigenvectors and eigenvalues of two matrices really mean? [closed]

Empirically I've noticed that diagonally dominant matrix G and it's diagonal version D (diagonal elements of G on the diagonal and all other elements are set to zero) produce similar eigenvalues and ...

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202
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### Quantification of the extent of periodicity in a time series using fractal analyses

I need metrics to quantify and compare the extent of periodicity between any two given time series, considering the time series were "almost periodic". By "almost periodic" I mean: if I were to take ...

5
votes

3
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1k
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### Looking for techniques of How to measure the Similarity/Dissimilarity between two images?

I would like to compute the similarity/dissimilarity between two images L and R.
I know one way which is : computing the histogram of blocks of each image, and then using Bhattacharyya measure I ...

2
votes

0
answers

240
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### Is there an universal (dis)similarity measure between two structures?

I'm always wondering is there an universal (dis)similarity measure
between two structures (let's say between two undirected simple
graphs)? I mean, not "the measure with universal parameter that we
...

13
votes

4
answers

24k
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### What is a good algorithm to measure similarity between two dynamic graphs?

I am using graphs to represent structure present in a scene. The vertices represent the objects in the scene and the edges represent the relationship between two nodes(touching, overlapping, none). ...

2
votes

2
answers

1k
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### Appropriate histogram comparison distance measure

I am working with hyperspectral image data in R, so I have subset an image to a region of 5000 pixels, each containing a vector 254 bands in length.
I would like to cluster this data in order to try ...

6
votes

2
answers

4k
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### similarity transformation into symmetric matrices

I want to determine some structures of matrices that can be transformed into a symmetric matrices using similarity transformation, i.e.,
$B=T^{-1}AT$
where $T$ is the similarity transformation ...

23
votes

0
answers

8k
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### An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?

A famous result in linear algebra is the following.
An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.
I know one proof using the Smith Normal Form (SNF). ...

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2
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1k
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### Similarity measure between 2 bi-partite graph.

Hello there, i need to solve this problem:
I have 2 different bi-partite weighted graph, g1 and g2 and i would like to measure their similarity, g1 and g2 may have different number of vertex and edges ...

1
vote

0
answers

191
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### Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis.
Correlation of the series changes over time and across ...

0
votes

1
answer

1k
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### How to use node similarity to measure subgraph similarity

For a semantic annotation task I am trying to calculate the semantic similarity between two sets of annotations: S1 and S2. Both sets consist out of multiple nodes from within one graph (in my case an ...

0
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1
answer

652
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### Fuzzy vector similarity

Hi all,
I have two multi-dimensional vectors representing documents $\vec{a}$ and $\vec{b}$.
Considering cases where there is no overlap between $a$ and $b$ ($a \cap b = \emptyset $), traditional ...

14
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0
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620
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### Some questions on unitarisability of discrete groups

In this post I would like to ask several of questions related to Dixmier problem. I will try to make the post as self-contained as possible.
A discrete group $G$ is unitarisable if for every Hilbert ...