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Questions tagged [hessian-matrix]

The Hessian matrix of function is used to second derivative test when $f$ has a critical point $x$. If the Hessian is positive definite at $x$, then $f$ attains a local minimum at $x$. If the Hessian is negative definite at $x$, then $f$ attains a local maximum at $x$. If the Hessian has both positive and negative eigenvalues then $x$ is a saddle point for $f$.

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Computing the Hessian from the Jacobian of the gradient

I'm trying to compute the Hessian of the function, $$f(X)=a^tX^2b$$ I used the perturbation approach to expand $f(X+H)$ as $$f(X+H)=a^t(X+H)b=\underbrace{a^tX^2b}_{f} + \underbrace{a^t(XH+HX)b}_{\...
Set's user avatar
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Sketchy use of multivariable chain rule under too weak hypoteses

I've found this statement in my real analysis course notes: Let $f: B_r (x_0) \subseteq \mathbb{R}^m \to \mathbb{R}$ ($B_r (x_0) = \{ x \in \mathbb{R}^m : d(x, x_0) < r \}$) be a function such ...
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Definition of covariant Hessian

I read the paper and on page two, it says that a linear function $A(t)$ is the covariant Hessian of a function $f$. Can anyone give me a definition of the covariant Hessian? I couldn't find anything ...
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Positive semi-definitiveness condition for multidimensional minimization case

We study the nonlinear problem \begin{equation} \underset{\mathbb{R}^2}{\text{min}}f(x) \end{equation} where $f(x)=x_1^2+x_2e^{x_1}-x_1x_2+x_2^2$ Evaluate whether the problem is convex. For a ...
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Least Squares Function Approximation and Convexity of Functions

I have been reading about Least Squares function approximation and am dealing with the following definition: Let $f$ be continuous on $[a,b]$ and let $W$ be a finite dimensional subspace of $C[a,b]$. ...
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This function has no saddle points: correctness of this reasoning

I would like to know if my reasoning is correct. I have the function $f(x, y) = e^{3x}(1+25x^2+25y^2)$ and I have to study the stationary points. After computing the gradient I found $$\begin{cases} 3 ...
Heidegger's user avatar
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What is the intepretation of a Hessian with only negative and zero eigenvalues?

I'm looking at a high-dimensional optimisation problem. Specifically, one involving a function $F(c_n)$ which depends on N parameters (in my case $N=200$). I have been able to minimise this function ...
anonymous2506's user avatar
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Double derivative of concave $F$ is negative? $(x-y)\cdot\nabla F(x)=0$ implies $\partial_{x-y}\partial_{x-y}F(x)<0$?

$F:\mathbb R^n\to\mathbb R$ is a strictly concave. $\nabla$ is gradient. $x,y,z\in\mathbb R^n$ Then, is it possible to prove that: $(x-y)\cdot\nabla F(x)=0$ implies $\partial_{x-y}\partial_{x-y}F(x)&...
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Block Matrix eigenvalues

I am working on a complicated optimization problem. I define $I\in \mathbb{N}, I > 1, n_I=\binom{I}{2}$. I try to optimize a scalar-valued function $\mathcal{O} : \mathbb{R}^n \longrightarrow \...
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Derivatives of multivariate Gaussian

I do not understand how to take the second derivative of the Gaussian: While I am certain that the result is correct, I do not understand how to get from the first to the second line of the second-...
Make42's user avatar
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Jacobian and Hessian of a vector

I am dealing with vector derivatives, and I am having a hard time generalising results. What I mean is that I can compute the results by hand, variable per variable, but I would like to do it in a ...
Babababa31's user avatar
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Does $x$ and $(\nabla f)(x)$ align as we approach the minima for a convex function?

Given a convex function $f(x): \mathbb R^n \to \mathbb R$, let $h(x)$ denote the cosine of the angle between $x$ and $(\nabla f)(x)$. $$h(x) := \frac{x^\top.(\nabla f)(x)}{\Vert x\Vert \Vert (\nabla f)...
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Levenberg-Marquardt algorithm and inverse Jacobian/Hessian

Let’s say I have a function $f(p,q):R^{n+m}→R$, with $p∈R^n$ and $q∈R^m$. I have a set of $q_{i=1,…,k},y_{i=1,…,k}$ and I want to find $p$ so I use the Levenberg-Marquardt algorithm to resolve the ...
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Morse function on a two sphere

For the past few days I've been studying the very basics of Morse theory and its connection to supersymmetric quantum mechanics. I'm following the lectures written by David Tong. To introduce the ...
luki luk's user avatar
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Relationship between hessian matrix and curvature [closed]

I am taking vector calculus this semester, and while researching about Hessian matrices for a project, I encountered this formula. enter image description here Could anyone explain how it is derived, ...
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Why $\nabla f$ do not exactly coincide with $D f$ (it's its transpose)

Is there any reason (historical, or of any other kind) to why $$\nabla f= \begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \\ \end{bmatrix}...
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The inverse of a specific case of symmetric matrix (scalar product of d dimensions vectors)

The problem is the following: For $i \in [N]$, let $v_i$ be a $1 \times d$ vector and $b_i$ a scalar. Moreover, let $A$ be a $N \times N$ matrix, whose (i, j)-entry is: $$ a_{ij} = \begin{cases} \...
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Matrix derivative of matrix commutator

I'm working on some functional and I have to calculate its second derivative with respect to some matrix-variables. I'm just left with the following derivative to perform: $$ V''= -6 i \lambda \gamma ...
Fredrigo6's user avatar
2 votes
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Find maximum function with 7 variables (REPOST)

My problem was not correctly stated in the old post. (Find maximum function with 7 variables) and because of multiple edits it confuses people. Here is the function I'm trying to find the maximum: $$f(...
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Inequalities of Hessian of distance functions on complete non-compact Riemannian Manifolds

I am interested in finding some inequalities relating the following expression with the curvature on a non-compact Riemannian Manifold $\frac{1}{d_1}Hess^{d_1} (\frac{\partial}{\partial x^\alpha},\...
Teo Rugina's user avatar
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Hessian of coordinate function on sphere

Denote by $S^n$ the unit sphere in $\mathbb{R}^{n+1}$, and consider the coordinate function $x_{n+1}$ on it, i.e. the function $(x_1, \ldots, x_{n+1}) \mapsto x_{n+1}$. Denoting by $\mathrm{Hess}(x_{n+...
AlexE's user avatar
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Hessian matrix determinant greater than zero in a saddle point?

I have the function $f:\mathbb{R}^3 \mapsto \mathbb{R}$ defined as $f(x, y, z) = xy+xz+yz+z-x$. I've calculated the Jacobian: $$ J_f = (y+z-1, x+z, x+y+1) $$ Which, by setting $J_f = \vec{0}$, reveals ...
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${\rm Hess}~r$ is scalar matrix $\implies$ $M$ is isometric to the space form

I'm trying to prove the rigidity part of a theorem in my paper, which requires the use of the classical Hessian comparison theorem's rigidity part: $${\rm Hess}~r=\frac{{\rm sn}_k'}{{\rm sn}_k}{\rm d}...
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Relation between Hessian of the squared distance function and the metric tensor

Let $(M, g)$ be a Riemannian manifold and let $d$ be the distance function associated with the metric tensor $g$. Define the function $$ f(x) := \frac12 d^2 (x,y) $$ for some arbitrary fixed $y \in M$....
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Classifying critical points when hessian has det0 [closed]

I am trying to find the critical points of $$f(x,y)=(x^2-y^4)(1-x^2-y^4).$$ There are 9, and I found all their coordinates. I’ve classified 6/9 using the hessian, but the remaining 3 have at least one ...
edster101's user avatar
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1 answer
71 views

Possible to have only zero eigenvalues of the Hessian of a harmonic function that is neither of the form $ax+by+cz+d$ nor a constant?

(Following an earlier post here) I intuit that if we restrict to functions $f(x,y,z)$ that are harmonic (i.e. satisfying $\nabla^2f=0)$ but neither of the form $ax+by+cz+d$ ($a,b,c,d\in\mathbb{R}$) ...
Solidification's user avatar
2 votes
1 answer
77 views

Proving nonexistence of local extrema of harmonic functions using Hessian

Edited I want to prove that solution to Laplace's equation do not support local maximum or minimum using Hessian. Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies ...
Solidification's user avatar
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1 answer
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Is this function convex over a set?

Well, I got a doubt: is this function $$f(x, y, z) = 2xy + yz$$ Convex over the set $$V =\{ (x,y,z) \in \mathbb\{R\}^3:\ x+y+2z \leq 1, x\geq 0, y \geq 0, z \geq 0\}$$ ? Being it a quadratic form, I ...
Heidegger's user avatar
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1 vote
2 answers
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Logit Gradient/Hessian derivations

I'm trying to follow the algebra leading from the gradient function to the Hessian in Logistic Regression, but I can't quite understand where I have gone wrong. I have the gradient function as: $$ \...
Jred's user avatar
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On checking the convexity of a multivariate function.

I want to check the convexity of the following multivariate function: $$K(Q, s) = h\frac{(Q-s)^2}{2Q}+b\frac{s^2}{2Q}+\hat{b}\frac{s\lambda}{Q}+K\frac{\lambda}{Q} +c\lambda,$$ where $Q>0$, $s \ge 0$...
Steven01123581321's user avatar
1 vote
1 answer
112 views

Is $f(x,y) = \dfrac{y+2}{x+y+2}$ convex anywhere? This is driving me crazy.

So, from Multivariate Calculus and Linear Algebra, I learned that: Given $f(x,y)$, I can find out if it is convex by checking if the eigenvalues of its Hessian are positive or zero (i.e. if the ...
William's user avatar
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3 answers
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On the Hessian of the Log-Determinant and the solution provided in Stephen Boyd's textbook

This is a follow up to another question on the second-order approximation to log-determinant in Boyd's textbook, the excerpt can be found here: Here, $f$ is the log-determinant, $f(Z) = \log(\det(Z))$...
Olórin's user avatar
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1 vote
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Bordered Hessian extremum verification

I have the following question: of course we know that in Lagrange multiplication method we can use bordered Hessian to check if some points are local extremum. But do this method, this bordered ...
mwrooo's user avatar
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3 votes
2 answers
65 views

How many positive elements in a diagonal matrix $D$ do we need to make $A+D$ positive definite for a real symmetric $A$ with $m$ negative eigenvalues

Consider a $n$-dimensional real symmetric matrix $A\in\mathbb{R}^{n\times n}$ having exactly $m$ negative eigenvalues and $n-m$ positive eigenvalues ($1\leq m\leq n$). Let $D\in\mathbb{R}^{n\times n}$ ...
Peng Yang's user avatar
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Does the Hessian correspond to the exterior derivative of the gradient 1-form? Or does its skew-symmetrization?

Question: Given a twice totally differentiable (not necessarily $C^2$) function $f: \mathbb{R}^m \to \mathbb{R}^n$, do its $n$ Hessian matrices correspond to the exterior derivatives of its $n$ ...
hasManyStupidQuestions's user avatar
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What does singular hessian in optimization tell me

I am doing optimization using maximum likelihood estimation, and when I am trying to get the standard errors of estimates using hessian matrix, I get non-invertible/singular hessian warning. After I ...
jasmine's user avatar
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How to prove that this polynomial function of $m$ variables is strictly convex?

Let $f:\mathbb{R}^m\to\mathbb{R}:=f(x_1,x_2,\dots x_m)={x_1}^{2k_1}{x_2}^{2k_2}\dots {x_m}^{2k_m}, k_i \in \mathbb{N}.$ Is there a way to show that $f$ is strictly convex? Clearly $0$ is the unique ...
Learning Math's user avatar
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How to determine a change of variable that makes the Hessian matrix diagonal?

Let say I have a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ that depends on two variables $\mathbf{x}=[x, y]$. The hessian matrix $H$ associated to $f$ $$ H(\mathbf{x}):= \begin{pmatrix} \...
duc4rm3's user avatar
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1 answer
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Understanding derivation of Hessian Sketch formula

I've been reading this paper and in equation 23 they ask us to consider the optimization problem. \begin{align} \hat{u} = \arg \min_{u \in C - x^t} \left\{ \frac{1}{2} \lVert Au \rVert^2_2 - \langle A^...
jeffj1355's user avatar
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What is the name of the middle matrix in a quadratic form $x'Ax$?

In a quadratic form, $x'Ax$, does matrix $A$ have a special name? I am trying to write a sentence that looks like "Using vector $x$ and matrix $A$ as the ______ matrix, we can write the quadratic ...
Adi's user avatar
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3 votes
1 answer
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Monotonicity of a multivariate function

Let $f:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$. Assume $f(x,y)$ is twise continuously differentiable and the derivative with respect to both arguments (i.e., $x$ and $y$) are monotone in ...
Andeanlll's user avatar
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A Hessian matrix (convex function) question. Given $u \in C^2(\mathbb{D}, \mathbb{R})$ s.t. $u(x) \geq \frac{1}{1-||x||^2}$.

Given $u \in C^2(\mathbb{D}, \mathbb{R})$ such that $$u(x) \geq \frac{1}{1-||x||^2}$$ And let $\Omega$ be the set of points where the hessian matrix, $Hu \geq 0$. Prove or disprove that $Du: \Omega \...
ben_xiaohai's user avatar
1 vote
1 answer
57 views

Differentiation involving vector variables

I am learning Lagrangian duality methods in an optimization course. Below is from my lecture note. Let $\ell(\textbf{x}; \lambda) =-x_1x_2+\lambda[(x_1-3)^2+x^2_2-5]$. Then, the Hessian of $\ell(\...
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General solution to the eigenvalue problem of a $3×3$ symmetric Hessian defining curvature

The symmetric Hessian for an implicit surface defined from a field variable $c(x,y,z)$ in Cartesian space is, $$ \nabla^2 c = \begin{bmatrix} c_{xx} & c_{xy} & c_{xz} \newline c_{xy} & c_{...
H.F.A.'s user avatar
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2 votes
1 answer
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How to calculate the principal curvature of a fitted surface by hessian matrix?

By searching relevant materials, I found a method to calculate the principal curvature of the fitted surface, which calculates the Hessian matrix, then obtains its eigenvalues, and finally directly ...
Bob's user avatar
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Classifying degenerate stationary points with the Hessian matrix

I've been wondering about the test to classify stationary points using the Hessian matrix. In the 2 variable case, things are fairly easy to deal with, I'm concerned with 3 or more variables. I've ...
TheInquisitiveOne's user avatar
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Eigenvalue Optimization, Find indefinite matrix with only one negative eigenvalue

I am new to the field of eigenvalue optimization. Say I have a symmetric matrices $A(x)\in\mathbb{R}^{2n\times2n}$ which depend on $x\in\mathbb{R}^n$. I now want to find situations in which one and ...
Hendrik's user avatar
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1 answer
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Is this multivariate function convex? And is a Hessian matrix of this form always convex?

I'm looking at the following function: $$K(Q, s) = h\frac{(Q-s)^2}{2Q}+b\frac{s^2}{2Q}+K\frac{\lambda}{Q}+c\lambda,$$ where $Q>0$, $s \ge 0$ and $h,b,c, \lambda, K \ge 0$ and are known. I want to ...
Steven01123581321's user avatar
4 votes
1 answer
225 views

Is Hessian with zero directions indefinite?

I have a Hessian matrix $\mathbf{H}$ of a function $f(\mathbf{x})$, evaluated at an extreme point $\mathbf{x}_0$. Lets assume $\mathbf{H}$ is non-singular. I can show that there exists a direction $\...
Ralf's user avatar
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Requirement of Hessian comparison theorem to prove that the squared distance function is strictly geodesically convex.

Let $(M,g), dim(M)=n,$ be a complete Riemannian manifold and let $f(x):=d(x,p)^2, p\in M$ being fixed. I'm interested in conditions where $f$ is strictly or weakly geodesically convex. I see the ...
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