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Questions tagged [descent]

Use for questions related to descent theory in topology.

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Find a descent direction at a saddle point

question: (Descent directions at stationary points). Assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is $C^2$ with $\nabla f(x) = 0$ and $\nabla^2f(x)$ indefinite (with both positive and ...
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Proposition 1.6.6, Etale Cohomology theory, Lei Fu

I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu. $\mathrm{Proposition} 1.6.6$ Let $g:S'\rightarrow S$ be a quasi-compact ...
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Descent data and trivialization of bundles via coherent isomorphisms of fibers

In this MO question I tried to understand how a trivialization of a bundle (continuous map) $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ is related to a coherent family of isomorphisms ...
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What is “distinguished cross section” (Solomon’s descent algebra proof)?

I’m attempting to read the proof of Solomon’s descent algebra from A Mackey Formula in the Group Ring of Coxeter Groups though I know nothing about Coxeter groups and Mackey formulas. I’m interested ...
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Non split real form of projective space

On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this ...
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Solving minimization problem with L1 norm using gradient descent

Im trying to solve the following minimization problem algorithmically to calculate X and E from Z by using gradien descent approach: minimization problem I am really struggling, how to take into ...
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Galois Descent for representations of finite groups OR a question about block matrices where the blocks are Galois conjugates

The question is quite long since I give some background but really I am interested in some very concrete fact about matrix representations. Scroll all the way to the bottom for a self contained ...
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Gradient of a sum log function

From Convex Optimizations Boyd and Vandenberghe, exercise 9.30, we have where $a_i^T$ is a row of the matrix A. $$ \min: f(x) = -\sum_{i=0}^m\log(1 - a_i^Tx) - \sum\log(1-x_i^2) $$ with variables $x$ ...
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The category of descent data

This is from Angelo Vistoli’s notes http://homepage.sns.it/vistoli/descent.pdf page $71$. Let $\mathcal{C}$ be a site and $\mathcal{U}=\{\sigma_i:U_i\rightarrow U\}$ be a covering of $U$. Let $\...
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Confusion with definition of foliation

Below is the definition of foliation of a manifold appearing in the book Introduction to Foliations and Lie Groupoids by Moerdijk and Mrčun. Definition 1. Let $M$ be a smooth manifold of dimension $...
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Examples of modules with non-effective descent data

Let me first summarise descent data for modules, following the Stacks project: Given a ring homomorphism $R \to A$ we can extend scalars for $R$-modules $V$ to get $A$ modules $A \otimes V$ ($\otimes ...
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Relation between seemingly distinct notions of Beck-Chevalley condition?

(Terminology of pseudofunctors and fibrations is mixed throughout.) The Beck-Chevalley condition is defined on the nlab as a property of a quadruple of functors: starting from an invertible 2-cell of ...
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(When) Is the category of descent data equivalent to the essential image of the base change?

Consider a family of continuous maps $(U_i\to U)$. For a family of bundles $(X_i\to U_i)$, TFAE: The family is a pullback of a single bundle $X\to U$; There exist transition isomorphisms satisfying ...
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Question about remark in Vistoli's notes on descent

Section 4.1.1 of Vistoli's Notes on Grothendieck topologies, fibered categories and descent theory motivates the formalism of stacks via topological spaces. Proposition 4.1 says that given an open ...
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Is the definition of fpqc topology in SGA 4.5 different from usual?

I am confused about Theorem 1.4.5, Example 1.6.4(b) in SGA 4.5, because here a covering of $U$ is a finite family $\{U_i\to U\}$ of flat morphisms such that $\coprod U_i\to U$ is surjective $\...
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Why does the cocycle condition on fiber isomorphisms imply a fiber bundle is trivial?

While reading about connections I stumbled on this entry of the encyclopedia of math. In the second paragraph of the comments section it's written that if a fiber bundle admits coherent isomorphisms ...
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Newton algorithm not finding minimum of $100(y-x^2)^2 + (1-x^2)$

Edit: Solved. I just forgot to update the inverse of the Hessian. I have updated the code now in case someone needs it. I am implementing some optimization algorithms (steepest descent, newton's ...
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Do the effective descent morphisms w.r.t the codomain fibration hint at the “right topology”?

An intuitive approach to basic descent theory for me started with open covers $ \left\{ U_i \right\}$, replaced them with a singleton covering $\coprod _iU_i\rightarrow X$, and then generalized to ...
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Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms'

I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter $\...
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Question about limit of cosimplicial diagram associated with a sheaf

Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
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Proposition 4.1 in Vistoli's notes on descent

Proposition 4.1 of Vistoli's notes says $\mathsf{Top}^2$ is a stack over $\mathsf{Top}$. The proof starts by looking at the fibered product $\coprod_iU_i \times _U\coprod_iU_i$, however this object ...
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Silly question about descent

Most sources say descent is defining an object over $S$ using objects over $U_i$ for some cover $\left\{ U_i \right\}$ of $S$. If I replace the covering family with a single arrow $\coprod _i U_i\...
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$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
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Show Direction of Steepest Descent is Unique

If f is a proper convex function and x is in the interior domain(f), how would one go about proving that the direction of steepest descent at x is unique? I intuitively get it, but don't get how one ...
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Asymptotic expansion using method of steepest descents

I am trying to find the first term in the asymptotic expansion of $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{1}{s^2}e^{t(s-m\sqrt{s^2-1})} ds $$ where $0<m<1$, $c<1$, as $t$ ...
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why use a small learning rate in gradient descent

I am new to neural networks and recently found out about gradient descent. Something does not sit right with me. x←x−λ∇fk(x) Why does this formula work? Wouldn'...
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Reference request: convergence property of continuous gradient descent?

Does anyone know of a text that treats the problem of gradient descent from a continuous perspective instead of a discretized perspective? For example, most text investigates the numerical properties ...
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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 $\{ d^\rho_1,\ldots,d^\rho_{k_\rho}\...
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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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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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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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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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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 f(x_k)}...
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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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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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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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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: $$\mathbf{x}_{n+1}=\...
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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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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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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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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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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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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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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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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 $5$-permutation:...
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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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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 activation:...
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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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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 Fréchet 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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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 ...