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9h
comment Can I solve a problem like a combination of PCA and compressed sensing?
You mention $E$, but it doesn't appear in your problem.
10h
comment What kind of geometry is useful to study for mathematical competitions?
The Art of Problem Solving book Introduction to Geometry is probably a good place to start. The book The Art and Craft of Problem Solving also has a chapter on geometry.
10h
comment Confusion over Lagrangian function
Yes, that's correct.
20h
comment Need ''The diameter of the prime graph of a finite group'' by Lucido
You could try asking on scholar.reddit.com.
20h
comment How to solve $\min \limits_{\mathbf{x}} \| \mathbf{Ax}-\mathbf{b} \|^2$?
Great answer. However I think the forward-backward method has been around a long time -- this paper by Tseng mentions that it was proposed (for monotone inclusion problems) in the 1979 Lions and Mercier paper.
1d
comment Confusion over Lagrangian function
Once you write the primal problem in a certain way, both the Lagrangian and the dual problem are specified. If that Lagrangian has a saddle point, then the optimal value for that corresponding dual problem is equal to the primal optimal value. (It would be interesting to think about whether a stronger statement could be made, along the lines you suggested, but we don't need a stronger statement.)
1d
comment Confusion over Lagrangian function
The Lagrangian (and the dual problem) depend on how the primal problem is written. If there are any constraints you don't want to dualize, you can include them in the $ x \in X $ constraint. One point of view is that you don't get a dual problem until you specify how to perturb the primal problem, and different ways of writing the primal problem give different ways to perturb it. Ekeland and Temam discusses the perturbation viewpoint, for example.
1d
comment Regarding Nesterov's smooth minimization
Check out Vandenberghe's 236c notes, such as the slides on smoothing and the slides on fast proximal gradient algorithms. Much of the literature has been digested and presented in a short, clear way. At the end of each chapter is a list of references, including to papers by nesterov.
2d
comment The Legendre-Fenchel transform of $BV$ semi-norm
A basic fact to be aware of is that (at least in a finite dimensional setting) the convex conjugate of a norm is the indicator function of the dual norm unit ball. Also, the conjugate of the conjugate is the original function (under mild assumptions), which helps to explain equation (2).
2d
comment Why memorize trig identities?
What are the 30 trig identities? Memorizing 30 sounds excessive to me, but as you said it's good to be able to derive them all (which you are able to do already).
Jun
26
comment When trying to learn analysis from bottom up, what numbers should I first construct?
Another option is just to start with a set of axioms for the real numbers.
Jun
26
comment The meaning of the symbol $\infty$ in Spivak's calculus book
I think he means that while he has defined $(a,\infty)$, for example, he has not given the symbol $\infty$ any independent meaning (and he doesn't need to). In fact, he could have equally well used some different notation for the interval $(a,\infty) $ such as $(a,\ldots) $.
Jun
25
comment What is the difference between only if and iff?
The mathematician R.L. Moore used "only if" to mean "if and only if". This sounds weird to us now, because it goes against the accepted convention, but I can see what Moore was thinking. The statement "A only if B" sounds like the statement "A if B", except that you are also given an extra piece of information: not just A if B, but A only if B.
Jun
18
comment What is topological learning?
I'm not sure, but maybe it means figuring out the structure of a graph. Just a guess.
Jun
12
comment How to show that Id + skew matrix is invertible
math.stackexchange.com/questions/1276665/…
Jun
11
comment Identical observations in linear regression
Often $ X $ is a skinny matrix and has a trivial nullspace, even if some of the rows of $ X $ are duplicates. What is the shape of $ X$ in your problem? Duplicate rows in $ X$ does not necessarily imply that $ X^T X $ is singular.
Jun
9
comment Can I do research just because I am able to?
I sympathize with your desire to live frugally and be free to do math research, but my impression is that without a PhD it will be very hard to find any job that pays you to do pure math research.
Jun
9
comment What is the latest work being done in the field of Mathematics? 6/8/2015
Recently there was very exciting progress on the twin primes conjecture that you can read about.
Jun
7
comment What are the desired questions that should to be in SM (stackexchangemath)?
You can google for what's on topic on math.stackexchange.
Jun
7
comment Visual representation of matrices
Sometimes it can be useful to view a large matrix as an image. Spy plots can also be useful to visualize the sparsity pattern of a matrix.