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Does every Young diagram have a unique minimal major index?

Given a Young diagram, $Y_\rho$, corresponding to an irreducible complex representation $\rho$ of the symmetric group $S_n$, we can associate a set of major indices $\{ ...
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32 views

Can every iterative algorithm be viewed as gradient descent over some objective?

In Algorithms for Non-negative Matrix Factorization, Lee and Seung give multiplicative algorithms derived from gradient descent on the Frobenius norm to find a non-negative matrix factorization. ...
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1answer
38 views

Is there a particularly simple example of geometric descent?

I'm looking for a particularly simple and familiar example of descent in geometry or topology in order to motivate the general definition. I'm not counting the definition of the arrow category ...
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22 views

Does faithfully flat descent work using restriction of scalars rather than extension?

Vistoli's notes on fibred categories and descent - http://homepage.sns.it/vistoli/descent.pdf - introduce (section 4.2.1) descent on modules over a commutative ring. The idea is as follows: ...
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1answer
29 views

Steepest Descent Sequence

How can I compute the first three iterates for the steepest descent sequence $f(x_1,x_2) = \frac{(x_1^2+3x_2^2)}{2}$ beginning at $x_0 = (\frac{\sqrt{3}}{2}, \frac{1}{2 \sqrt{3}})^T$ $\nabla ...
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11 views

Increase rate of convergence for steepest descent

How can one find a transformation matrix $T$ for $y=Tx$ that decreases the condition number of the Hessian of a quadratic function and decreases the iteration time?
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49 views

Unwinding descent via Barr-Beck

Let $f: U \rightarrow X$ be a faithfully flat morphism of nice schemes (quasiseparated, quasicompact, and anything else I might have forgotten). One can understand descent in quasicoherent sheaves ...
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89 views

Derivation of Steepest Descent Direction used in Line Search Methods

In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$ This resulted in $$ p_k=-\frac{\nabla ...
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2answers
42 views

Gradient decent using Taylor Series

I'm reading a book about Gradient methods right now, where the author is using a Taylor series to explain/derive an equation. $$ \mathbf x_a = \mathbf x - \alpha \mathbf{ \nabla f } (\mathbf x ) $$ ...
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1answer
87 views

How does a section of a stack give a sheaf?

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a ...
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0answers
40 views

When is the map “attaching irreducible components” an effective isomorphism?

Let $\mathcal C$ be a category with fiber products. We say that a morphism $X \to Y$ in $\mathcal C$ is an effective epimorphism if the sequence of sets $$ \text{Hom}(Y,S) \to \text{Hom}(X,S) ...
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24 views

Does the method of steepest decent always move in an orthogonal direction between iterations?

I understand everything, I think, about the method but the result (or requirement) that successive steps are orthogonal to each other. SO, with the formula for this algorithm as: ...
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39 views

G-equivariant invertible sheaves on affine curves

Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism ...
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38 views

As a beginner, I would like to solve convex quadratic maximization problem with a gradient descent variant in probability simplex?

I know the basics of gradient approaches to optimize the function iteratively, but for this case have have a equality constraint as $\sum_{i=1}^Nx_i = 1$ where each $x_i \geq 0$ with the objective ...
3
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1answer
190 views

SGA 4.5 proof of Hilbert 90 and semilinear Galois action

In SGA 4.5's proof of Hilbert 90, proposition 1.5.2(that the inclusion $V'^G \otimes_k k' \rightarrow V'$ is an isomorphism) is deduced from faithfully flat descent as stated in 1.4.5. The way that I ...
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1answer
156 views

Using faithfully flat descent to prove representability of a functor in a simple case

Let $k$ be a field with a fixed separable closure $k_s$ and $G$ a finite type $k$-group scheme. Assume $F:(\mathrm{Sch}/k)^{opp}\rightarrow\mathrm{Set}$ is a contravariant functor whose restriction ...
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0answers
41 views

Optimization, descent direction, neccessary condition

I'm learning about nonlinear, unconstrained optimization. In my book it says that a descent direction $p_k$ must satisfy: $$p_k\nabla f(x_k)^T < 0$$ This seems to mean that $p_k$ must be obtuse to ...
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1answer
71 views

What is the difference between projected gradient descent and ordinary gradient descent?

I just read about projected gradient descent but I did not see the intuition to use Projected one instead of normal gradient descent. Would you tell me the reason and preferable situations of ...
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132 views

Intuition about multiplicative gradient descent

Suppose we want to minimize a function $f(x)$ wrt $x$, i.e., we want to solve, $$x^* = \arg \min_x f(x)$$ One method to solve such problems is gradient descent. In gradient descent, one uses the ...
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1answer
142 views

Can I check smoothness after a base-change

Let $X\to S$ be a flat morphism of noetherian schemes. I know that I can check smoothness on the geometric fibers to see whether $X\to S$ is smooth. Let $T\to S$ be a surjective morphism. Under what ...
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1answer
238 views

Machine Learning, why not use matrix multiplication instead of gradient descent?

If we want to minimize our Cost function for a given set of data, why do we use gradient descent and continually guess values until we find a min value for theta when when can just use matrix ...
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2answers
228 views

combinatorial descents finding the number of permutations with criteria

I need help with the following: Define a descent of a permutation to be $j$ when $p_{j+1} < p_j$. Then the descent set of a permutation is the set of all descents. For example, the ...
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1answer
264 views

Finding descent direction of quadratic function

I have a quadratic function: $f(x) = 24x_1+14x_2+x_1x_2$ and point $x_0 = (2,10)^T$ with $f(x_0) = 208$ And the first question is "give descent direction r in $x_0$" The second question "is f convex ...
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0answers
66 views

Rate of convergence of a single-neuron Perceptron network

I'm implementing a Perceptron network which basically consists of a single neuron in a single layer, trying to learn an OR logic port (linearly separable), but using the sigmoid function as ...
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2answers
416 views

Solving Linearly Constrained Quadratic Programming with Coordinate Descent

Does anybody have any idea about how to solve the following problem with Coordinate Descent? \begin{align} \min &\quad \mathbf{x}^{\top}P\mathbf{x} + b^{\top}\mathbf{x}\\ \text{Subject to}& ...
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1answer
140 views

Fréchet mean between points in $\mathbb{R}^3$

Let $X$ be a set of $n$ points in $\mathbb{R}^3$ and $f_m$ be the Fr├ęchet mean, i.e.: $$ f_m= \arg \min_{p \in M} \sum_{i=1}^n w_id^2(p,x_i) $$ where $(\mathbb{R}^3,d)$ is a complete metric space, ...
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1answer
543 views

How to prove the Energy function of a Hopfield net is monotonically decreasing?

How to prove the Energy function of a Hopfield net is monotonically decreasing? $E = -1/2 \sum_{i,j} {w_{ij}}{s_i}{s_j} + \sum_{i}^{}s_{i} \theta_i$ I'll assume a proof involves the standard ...
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199 views

Do cokernels in RingSpc automatically lead to descent?

I'm currently interested in the following result: Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" ...