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6
votes
0answers
123 views

Does the Mandelbrot set have infinite VC dimension?

Define a binary classifier for points in the complex plane, whose parameter $\theta$ is an isometry of $\mathbb{C}$, and which classifies $z \in \mathbb{C}$ based on whether or not $\theta(z)$ is in ...
2
votes
0answers
55 views

Maximum-likelihood estimation for univariate responses from multivariate data

I am new in the field of machine learning, so I hope I will be able to formulate my question in a clear way... I have some data represented by vectors $\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n ...
7
votes
1answer
168 views

“Separated” version of Sauer's lemma on VC classes

Sauer's lemma, a well-known result in computational complexity theory, learning theory, and combinatorics, states the following: Let $\Phi$ be a collection of subsets of a set $U$, and assume that ...
2
votes
1answer
164 views

Epsilon-approximations of set systems with finite VC-dimension

ECorollary 6.9 in A Guide to NIP theories by Pierre Simon proves the following Theorem. For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ ...
0
votes
0answers
47 views

Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...
0
votes
0answers
34 views

Is it possible to distinguish between to edge orientation while learning a network structure?

I'm considering the case of learning bayesian network structure using a dataset $\mathcal{D}$ with scoring methods : $$\mathcal{G}^*=\max_{\mathcal{G}}\text{Score}(\mathcal{G}, \mathcal{D})$$ I'm ...
5
votes
1answer
107 views

assumptions on local rademacher complexities

A lot of the work on Local Rademacher complexities of Koltchinskii, and Bartlett for fast rates of convergence is based on Bousquet's version of Talagrand's inequality [1] (Theorem 2.11). However the ...
-2
votes
1answer
436 views

AI / Machine Learning related to high/modern/front mathematics [closed]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
5
votes
1answer
418 views

Is there a mistake in Vapnik's “Basic Lemma”?

I have a concern about the "Basic Lemma" which Valdimir Vapnik states and proves in his 1998 book Statistical Learning Theory (ch. 14.3, pp. 574–76): It seems like a certain coefficient should have ...
1
vote
2answers
267 views

A machine learning application question

I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve. I want to predict the nature of user activity on a ...
0
votes
1answer
140 views

What is the Bahadur-Anderson Algorithm?

What is the Bahadur-Anderson Algorithm, and which book could one read to learn it?
1
vote
3answers
286 views

A sampling and learning question

Suppose there is an oracle that returns a number $b \in \mathbb{Z}_{n}$ whenever I press the button. We have $b = a + e$, where $a \in \mathbb{Z}_n$ is a fixed number and $e$ is sampled according to ...
1
vote
0answers
69 views

Vertex cover for hamming graphs representing sets of bounded VC dimension

Let $S$ be a set of binary vectors (in $\lbrace 0,1 \rbrace^m $) whose VC dimension is $d$. Let $H$ be the Hamming graph generated from this set where each node represents a binary vector and two ...
1
vote
0answers
212 views

VC dimension and boolean hypercube subgraphs

Are there any well studied graph theoretic properties that are common to all subgraphs of the boolean hypercubes that have a given VC dimension d.
3
votes
2answers
548 views

Vapnik-Chervonenkis dimension of lines in the plane

I'm having some problems with this problem concerning VC dimensions ( http://en.wikipedia.org/wiki/VC_dimension ), I hope for some helping input. Given a set $L$ of $n$ lines in the plane, define a ...
4
votes
1answer
1k views

Monotonicity of the hard EM algorithm.

Consider the problem where we want to find a maximum likelihood estimate of $\theta$, given $X$ and $$P_\theta(Y) = \sum_z P_\theta(Y,x)$$ where $x$ is a latent variable. I know that the soft EM ...
2
votes
0answers
655 views

Classical Multidimensional Scaling

Hi, I am doing an MDS with a distance matrix coming from geodesic distances between points X on a 3d mesh (ie., not euclidean distances), and try to find points Y in euclidean space which best ...
8
votes
4answers
2k views

Reference request for manifold learning

I am interested in learning about manifold learning (no pun intended) and would like to know of some references that discuss the subject from a more geometric perspective. By manifold learning I mean ...
1
vote
5answers
3k views

Nodes clusters with a distance matrix

Hi, I have a (symmetric) matrix $M$ that represents the distance between each pair of nodes. For example, A B C D E F G H I J K L A 0 20 20 20 40 60 60 60 100 120 ...
2
votes
3answers
6k views

The Polynomial Kernel

I Have seen two versions of the Polynomial Kernel during my time learning Kernel Methods for things such as regression analysis. 1) $\kappa_d(x,y) = (x \cdot y)^d$ 2) $\kappa_d(x,y) = (x \cdot y + ...
2
votes
1answer
93 views

Ranking sources at variable(random) frequencies

Hi, I have this math modeling problem that I need help with. If I have 3 data sources, each being updated at different frequencies, what would be the best way to rank them so the less frequent ...
10
votes
3answers
333 views

disconnected or poorly connected graphs in sport ratings systems

I've briefly read about rating systems that provide rankings to players based only on their performance wrt other players, in the context of chess. (for example, elo). When there is a lot of ...