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12h
comment Convex Optimization Closed Form Solution
@dohmatob hmm, I'm probably misunderstanding something. Let's suppose $\gamma > 1$. Let $\hat{x}$ be the optimal point you described, and let $x$ be the point I described. (So $x$ has one non-zero component, whereas $\hat x$ has $\gamma$ non-zero components.) Then isn't it true that $\| x \|_1 = \| \hat x \|_1$ and also $s^T x = s^T \hat x$? So it seems like both points are optimal, unless I'm mistaken.
17h
comment Convex Optimization Closed Form Solution
@dohmatob Thanks, I just made an edit to address that.
17h
revised Convex Optimization Closed Form Solution
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1d
comment How to understand mathematics on a deep level?
For some reason math authors often make zero effort to motivate the material or explain how someone might have thought of it. They might just throw out some very abstract definitions with no commentary. It helps me when I find books that attempt to explain the motivation and intuition.
1d
revised Convex Optimization Closed Form Solution
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1d
comment How is the Fourier transform a geneeralization the the Fourier series?
One way to motivate the Fourier transform (in its various incarnations) is to seek a basis of eigenvectors (or eigenfunctions) for the shift operator (in various contexts).
2d
answered Convex Optimization Closed Form Solution
2d
comment Convex Optimization Closed Form Solution
Nice explanation. I think we need to worry about the possibility that the half space already contains the origin, in which case the solution is just $ x =0$.
Jul
25
comment Convex Optimization Closed Form Solution
You can take a simple visual approach to solving a). Of all points in the given half space, which one is closest to the origin? Starting from the origin, the quickest way to reach the half space is to move in the direction $ s $.
Jul
25
comment How to overcome the temptation to read many books covering the same topics
I think it's good to use multiple textbooks when learning a subject. They each have their own strengths and weaknesses, and their own approaches. Some explain certain topics more clearly than others. You can read several of the best books simultaneously.
Jul
24
comment How to help a postgraduate student to write a book
There are many ways to help someone write a book, for example you could proofread it for them. Without more details about the situation it's hard to know what a helpful answer would be to this question.
Jul
24
comment What is the precise definition of 'between'?
I don't think there is a completely standard definition, but hopefully the author makes the precise meaning clear in context.
Jul
24
answered Why does the Hessian work?
Jul
22
comment Why is periodic harmonic analysis only possible with sines?
I might not know the history correctly, but I think perhaps Fourier was looking for solutions to the heat equation, and guessed that there should be separable solutions, and plugging in solutions of this separable form he found solutions involving sine and cosine (and sums thereof). He then made the leap (or so I was told) of guessing that "any" function could be written as a linear combination of sines and cosines, which would allow him to solve the heat equation for arbitrary initial values. A completely different viewpoint is to search for eigenfunctions of the shift operator.
Jul
22
comment Books on multivariable calculus
I wrote a short set of notes (15 pages) called "Quick Calculus" that attempts to derive the main results of calculus very quickly, using intuition rather than rigor. You could try taking a look.
Jul
22
comment Rudin's equivalent in Linear Algebra
This is an awesome book, and the proof (using annihilators) that $ A $ and $ A^T $ have the same rank is a great example of how an abstract approach can be the most enlightening.
Jul
21
answered Rudin's equivalent in Linear Algebra
Jul
21
comment Example of a surface where more than one coordinate patch is needed.
What software was used to make this picture?
Jul
21
comment What is the difference between stationary point and critical point in Calculus?
According to some authors at least, a critical point is a point where either $f'(x) = 0$ or $f$ is not differentiable, whereas a stationary point is a point where $f$ is differentiable and $f'(x) = 0$. See mathworld.wolfram.com/CriticalPoint.html and mathworld.wolfram.com/StationaryPoint.html for example.
Jul
20
answered Vector Norm addition