Questions tagged [duality]

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Recover Primal Linear Programming Solution from Dual with LAD Regression?

This link discusses different ways of writing a classic LAD regression with a linear program. The classic way of writing LAD regression ($y = X \beta + r$) as a linear program is \begin{equation} \...
2 votes
0 answers
45 views

LASSO and duality theorem

I am confused with Lagrange duality theorem. Let us consider the problem $$ \hat{\beta} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} \left[\sum_{i=1}^{n}(y_{i} - \beta_{i})^{2} + \lambda \sum_{i=...
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1 vote
1 answer
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Demonstration and Interpretation between a Fisher matrix and its dual space which is covariance matrix

I have a simple (maybe not) issue about the interpretation of the link between Fisher information matrix and its inverse which is the covariance matrix. How to formulate that a line of Covariance ...
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Primal Problem of SVM

The primal problem of SVM is denoted as below. $$min_{w,b}\left(\phi \:\left(w\right)\right)=min_{w,b}\left(\frac{1}{2}w^Tw\right)$$ Subject to $$y_n\left(w^Tx_n+b\right)\ge 1,\:n=1,2,3,...,l$$ And If ...
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What are the Karush–Kuhn–Tucker conditions for $\min_x \frac{1}{2} ||x-u||_2^2:$ subject to $||x||_1\le c$

What are the Karush–Kuhn–Tucker conditions for $\min_x \frac{1}{2} ||x-u||_2^2:$ subject to $||x||_1\le c$ Apparently these are the conditions but it's not all that clear how this can be applied to ...
2 votes
0 answers
54 views

What is the intuition of a dual?

I have been hearing that the Ridge regression is the dual to the GP (Gaussian process regression). What does this mean? Can someone please give an intuition on what 'dual' is. My impression of the '...
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0 votes
1 answer
52 views

Why is the Dual Formulation a valid reparametrization of a regression model

In polynomial regression problems, in which an input vector $\underline{\phi}(\underline{x})$ is used to map a feature vector to a higher dimensional space (an example of this being $(x_{1}, x_{2}) \...
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1 vote
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36 views

A PCA related problem [closed]

Consider the problem of shape averaging. In particular, suppose $X_i\ (i = 1,\dots, M$) are the input matrices, $X_i\in \mathbb R^{N\times2}$, with each sampled from corresponding 2D positions of ...
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2 votes
2 answers
113 views

How to solve MNP (minimum norm) problem in SVM?

I'm reading an article, which says that MNP (minimum norm problem) can be solved as SVM. In the minimum norm problem, we're given a set of points in $R^d$ and need to find a point in convex hull of ...
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2 votes
1 answer
122 views

Quadratic programming and interpretation of dual solution (Lagrangian)

Note: this question is about a common data science problem, but I am solving it using a specific piece of software. I believe the problem is common enough that these principles will be common across ...
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1 vote
0 answers
73 views

Primal solution exists but dual does not

I am working on the follwoing nonlinear model. Min z=10(1-$\exp$(−3x) ) subject to: x $\leq$ 3 When I solve this problem on LINGO, I got the message "dual solution does not exist but primal ...
1 vote
1 answer
376 views

How to recover primal problem from its dual counterpart

I am asking this from context of optimization in machine learning. We often talk about a primal problem and how this primal problem can be solved by first converting it into a dual problem (Using ...
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7 votes
1 answer
2k views

Earth Movers Distance and Maximum Mean Discrepency

By Kantorovich-Rubinstein duality the Earth Movers Distance (EMD)/Wasserstein Metric is equivalent to Maximum Mean Discrepancy (MMD) correct? See here for a more thorough explanation. Why then does ...
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