Questions tagged [hessian-matrix]

Question 1

Consider a self-concordant function $f(\cdot)$. This implies; $$ \|f'''(x)[u]\| \leq M \|u\|_{f''(x)}^{3/2} \; \text{for some M and} \; \forall u \in \mathbb{R}^{n}, \text{and} \; x \in \text{domain} \...

Question 2

If $f$ and $g$ are $C^2$ functions $\mathbb{R}^2\to\mathbb{R}$ having the same zero set $\cal C$. I want to ask whether $$\kappa_f:= \frac{\text{Hess}_f(t,t)}{\|\nabla f\|}$$ is equal to (up to a sign)...

Question 3

$\def\diag{\operatorname{diag}} \Lambda^{-1}=\diag\left(\dfrac{1}{\lambda_i}\right).$ $\lambda_i \geq 0$ $w_i \in \mathbb{R}$ $C>0, C \in \mathbb{R}_{++}$ $\mathbf w$ is length $d$ and so is lambda....

Question 4

Page 147, equation (3.9.9, of "The Finite Element Method Linear Static and Dynamic Finite Element Analysis" by Thomas J. R. Hughes contains the following formula $$ \begin{bmatrix} \dfrac{\...

Question 5

For which triples $(l,s,h) \in \mathbf{Z}^3_{\geq 0}$ does there exist an example of a smooth function $f\colon \mathbf{R}^2 \to \mathbf{R}$ that has $l$ local minimums (low points), $s$ saddle points,...

Question 6

For a smooth multivariable function $f \colon \mathbf{R}^2 \to \mathbf{R}$, a critical point of $f$ is any $(a,b)$ at which $\nabla f(a,b) = \mathbf{0}$. Whether a critical point corresponds to a ...

Question 7

I am studying the convexity properties of the negative log-likelihood in multinomial logistic regression. Let me briefly set up the notation: We have a dataset $$ D = \{(x_n, y_n)\}_{n=1}^N, \quad ...

Question 8

In Leonard Dickson's Modern Algebraic Theories (p. 3), the Hessian $h$ of a function $f$ is introduced, where the elements of the ith row are $\frac{\partial^2}{\partial x_i \partial x_1}, \frac{\...

Question 9

The mixed partial derivative $\frac{\partial^{2} f}{\partial x \, \partial y} := \frac{\partial}{\partial x}\!\left( \frac{\partial f}{\partial y} \right)$ is obtained by first differentiating with ...

Question 10

I have a planar complex projective cubic, let’s call it $F$. I’ve proven that it’s nonsingular and I’m now asked to prove that $D=\det(H(F))$ is again a smooth cubic. ($H$ is the hessian matrix of $F$....

Question 11

Let $f\colon \mathbb{R}^d \to \mathbb{R}$ be twice continuously differentiable. In the theory of Hamilton–Jacobi–Bellman or convex analysis in general one can encounter conditions on data like $$(\...

Question 12

Setup to the problem: We are going to determine the stationary points of the function $5x^3 - 3yx - 6y^3 - 2$ in the region $-1 \leq x \leq 1, \ -1 \leq y \leq 1$. Calculate the gradient $\nabla f(\...

Question 13

This proof was taken from the book Multivariable Mathematics by Theodore Shifrin. How does the definition of continuity in the second part of the proof connect to the claim that $$f(\mathbf a+\mathbf ...

Question 14

Problem formulation I would like to generalize the following well-known and super-nice formulas for the gradient $\nabla J(x)$ and Hessian $\nabla^2 J(x)$ of a quadratic cost $J(x)\triangleq x'Qx$ \...

Question 15

Let $\mathcal{l}$ be the following quadratic loss function: $\frac{1}{2} \theta ^t H \theta$ where H is Hessian matrix, $\theta \in R^d$ is the parameter vector. If we define the regularized version ...

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Questions tagged [hessian-matrix]

Properties of self-concordant functions

Two functions having the same zero set

If the second partial derivatives are matrices, can I plug them into the a new matrix and call new matrix the Hessian $H$ to show $H \succeq 0$?

The meaning of the elements of the inverse of the Jacobian (and Hessian)

Is there a smooth function having any given number of extrema and saddles?

Is there vocabulary for critical points of multivariable functions at which the second partial derivative test fails?

Why is the Hessian of the negative log-likelihood in multinomial logistic regression positive definite?

Why does the product of the Hessian and the transformation determinant only involve first derivatives?

Index notation for mixed partial derivatives – $f_{xy}=(f_x)_y$ or $f_{xy}=(f_y)_x$?

Is the hessian of a smooth planar projective complex cubic again smooth?

Characterization of positive definite matrices and convexity

How do I find and classify a stationary point of the function $5x^3 - 3yx - 6y^3 - 2$ using Newton's method and the Hessian eigenvalues?

How does this proof connect to the claim?

Gradient and Hessian of a pseudo-quadratic form

Hessian matrix and gradient vector for the regularized quadratic function

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