3
votes
2answers
41 views

If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. [on hold]

If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. I do not know how to approach this problem and would appreciate advice how to proceed.
3
votes
2answers
4k views

Prove that the min and max of 2 continuous function are continuous

Prove that if $f$ and $g$ are continuous functions the so are min⁡{f(x),g(x)} and max⁡{f(x),g(x)} I know this is true when $f$ and $g$ are not intersect each other, then I can compare them. However, ...
4
votes
2answers
72 views

Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
2
votes
1answer
25 views

Representation of an invertible square matrix as a product of elementary matrices.

Suppose $A$ is an invertible $2 \times 2$ matrix. What is the smallest integer $n$ such that $A$ is a product of $n$ elementary matrices? My guess is that at most 4 elementary matrices are ...
1
vote
0answers
27 views

Close-form solution for distribution of the stopping time for a path-dependent random process?

A time series $\{x_s\}_{s=1}^{t}$ is generated from $N(\bar{x},1/b)\ i.i.d.$. Parameter $\bar{x}$ is unknown, but we have a prior distribution $N(\phi_0,1/a)$ for it. Define conditional expectation ...
1
vote
2answers
31 views

Dirichlet inverse of $(-1)^n$

I was tinkering around and noticed the Dirichlet inverse of $\,f(n) = (-1)^n$ seems to be $$ f^{-1}(n) = -\mu\!\left(n\,/\,2^{\nu_2(n)}\right)\left\lceil 2^{\nu_2(n)-1} \right\rceil, $$ where $\nu_p(n)...
2
votes
2answers
57 views

Evaluating the sequence

I am currently working on the problem. Find $$\sum_{n=1}^{\infty} \frac{x_{n}}{n+2}$$ when $x_{n+2}=x_{n+1}-\frac{1}{2}x_{n}$ with $x_{0}=2$, $x_{1}=1$. I was able to find $x_{n}$ to be $x_{n}=(\...
0
votes
4answers
34 views

Formula for factorization of a Quadratic Equation?

To be clear I am looking for an equation to go from $$Ax^2 + Bx + C = 0$$ To $$(Dx + E)(Fx + G) = 0$$ And I need it to be able to be done in a computer as it will be going in my app. Thanks in ...
2
votes
1answer
217 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 ...
1
vote
1answer
92 views

Difference between Gradient Descent method and Steepest Descent

What is the difference between Gradient Descent method and Steepest Descent methods? In this book, they have come under different sections: http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf ...
1
vote
0answers
48 views

Gradient descent: L2 norm regularization

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be: $ w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t) $ $p(\mathbf{y} = 1 | \...
0
votes
0answers
16 views

Gradient descent with multiplication term

Say I have the objective: $\arg \min_{R, T} \|y - RTx\|^2_2$ where, R and T are matrices (not necessarily square) and y and x are known vectors. I wish to try and optimize R and T using Gradient ...
6
votes
2answers
1k views

Gradient descent with inequality constraints

Suppose we are given a convex function $f(\cdot)$ on $[0,1]$. One wants to solve the following optimization problem: \begin{equation} \begin{aligned} & \text{minimize} && \sum_{i=1}^n \...
2
votes
1answer
58 views

Lipschitz Number in Gradient Descent

During gradient descent, if an objective function's value is greater than the previous iteration, would use of an orthogonal vector to the update vector be advantageous? Regarding trust regions, the ...
0
votes
1answer
26 views

Interpretation of points in covering spaces as homotopy classes of paths [on hold]

If $p:\widetilde{X} \to X$ is a covering map, $y \in \widetilde{X}$ determines a homotopy class of paths in $X$ joining the base point $x_0$ to the point $p(y)$. But a homotopy class of paths in $X$ ...
0
votes
1answer
41 views

Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
12
votes
1answer
446 views

Stochastic gradient descent for convex optimization

What happens if a convex objective is optimized by stochastic gradient descent? Is a global solution achieved?
2
votes
1answer
26 views

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to?

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to- $\frac{\Phi(1)-\frac{1}{2}}{\Phi(2)-\frac{1}{2}}$ $\frac{\Phi(1)+\frac{1}{2}}{\Phi(2)+\frac{1}{2}}$...
0
votes
1answer
148 views

Gradient descent for periodic function

Problem: minimize E=$\sum_{t=0}^T [ Y(t)-\sum_{k=0}^K(X(t+k)*cos(k*F+PHI)) ]^2$ where F, PHI - has to be optimized; Y(t), X(t) 1D arrays are given; t=0...T; T=1000; k=0..K; K=10; I can use FFT ...
0
votes
1answer
47 views

Convergence of steepest gradient descent

The description of gradient descent in Wikipedia says: $$x_{n+1} = x_n - \gamma_n\nabla F(x_n)$$ for $n = 0,1,2,...$ Suppose that $x_n$ converges to $x$. Then, is it always true that $\nabla F(x) = ...
4
votes
1answer
983 views

Gauss-Newton versus gradient descent

I would like to ask first if the second order gradient descent method is the same as the Gauss-Newton method. There is something I didn't understand. I read that with the Newton's method the step we ...
1
vote
1answer
18 views

Implementing gradient descent based on formula

The gradient descent algorithm is given as : repeat { $$\displaystyle \theta_j := \theta_j - \frac{1}{m} \alpha \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)}) x^{(i)}_j $$ } Given these values : <...
3
votes
2answers
40 views

How to precisely define a function that chooses randomly from a finite set?

Let $A = \{1, 2, \ldots, n\}$. I want to define a function that picks with uniform probability an element in $A$, so that $$f(A) = i \in A.$$ I don't know how to precisely define this mathematical ...
2
votes
1answer
66 views

A Variant of Gradient Descent

Suppose I have some objective function $f(\beta)$ which I would like to minimize for $\beta$. A standard gradient descent would be $\beta^{(t+1)}=\beta^{(t)}-\alpha \nabla f(\beta^{(t)})$, where $\...
2
votes
0answers
71 views

Gradient descent via polynomial approximation

It seems that most proofs of convergence for gradient descent algorithms rely on strong conditions on the first and second derivatives of the function, for instance that $$|f''(x)| \leq K$$ over the ...
0
votes
1answer
26 views

Fourier transform property(uniformly converges) proof

Suppose that f is a 2π-periodic function that satisfies the estimate \begin{equation} |f(x)-f(y)|\leqslant M|x-y|^\alpha \end{equation} for an 0< $\alpha$ <1 Show that $S_n(x)$ converges ...
1
vote
3answers
304 views

Michael Spivak's Calculus - Chapter 1, Problem 19

Problem 19. The fact that ${a^2}\ge{0}$ for all the numbers a, elementary as it may seem, is nevertheless the fundamental idea upon which most important inequalities are ultimately based. The great-...
2
votes
0answers
63 views

Accelerated gradient descent versus nonlinear conjugate gradient descent

Let's consider smooth and convex minimization problem, i.e. $min f(x)$, where $f$ is not necessarily a quadratic function. If measured by iterations, Accelerated Gradient Descend (AGD) has $O(1/T^2)...
2
votes
2answers
379 views

Gradient descent vs ternary search

Consider a strictly convex function $f: [0; 1]^n \rightarrow \mathbb{R}$. The question is why people (especially experts in machine learning) use gradient descent in order to find a global minimum of ...
1
vote
1answer
125 views

A constrained gradient descent algorithm

I am looking for a way to find a solution to the constrained minimization problem using the gradient descent Algorithm. it follows ...
2
votes
2answers
30 views

Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
1
vote
1answer
71 views

Time bound for gradient descent

Have you seen any analytic bound on gradient descent (for number of iterations to achieve to $\epsilon$ error, and possibly based on the form of cost function and initial value)? Here is the problem; ...
16
votes
4answers
9k views

Gradient descent with constraints

In order to find the local minima of a scalar function $p(x), x\in \mathbb{R}^3$, I know we can use the gradient descent method: $$x_{k+1}=x_k-\alpha_k \nabla_xp(x)$$ where $\alpha_k$ is the step size ...
3
votes
3answers
2k views

Why does gradient descent work?

On Wikipedia, this is the following description of gradient descent: Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
1
vote
0answers
16 views

Is it possible to solve for $x$ using the lambert W function in the expression ${\ln\left(x\right)}=(t-x)^2$?

${\ln\left(x\right)}=(t-x)^2$ $\pm\sqrt{\ln\left(x\right)}+x=t$ $\mathrm{e}^{\sqrt{\ln\left(x\right)}+x}=e^t$ And that is as close as I can get it to the form $x\mathrm{e}^x$. What do I do next? ...
1
vote
1answer
77 views

why value of Trigonometric ratios of angle and its reference angle are same?

I'm learning Trigonometry right now with myself and at current about how to find trigonometry ratios of angles greater than $90^\circ$. I came to know that for finding trigonometric ratio of these ...
0
votes
0answers
105 views

Multivariable Gradient Descent

If I have a function $S: \mathbb{R}^2 \to \mathbb{R}$ that describes energy falloff in space. I have a source $S$ positioned at $(S_x, S_y)$ and the intensity at any given point in space (x, y) is $...
0
votes
1answer
397 views

Gradient descent rule

In machine learning a very common technique to use as a training algorithm (in NN) is the gradient descent rule. I understand that it is an iterative process of increasing each of the weights based on ...
0
votes
0answers
14 views

Scaled gradient descent

Consider the unconstrained minimization \begin{align*} \min_{x\in\mathbb{R}^n}f(x) \end{align*} One iterative approach to obtaining a solution is to use the gradient descent algorithm. This algorithm ...
0
votes
0answers
32 views

Gradient descent algorithm

Hello I'm trying to understand how the Gradient Descent Algorithm works. There is a formula that I found on wikipedia and that I cannot justify: https://www.wikiwand.com/en/Gradient_descent#/...
2
votes
1answer
288 views

Minimization and gradient descent

I am bit puzzled by using gradient descent method. My doubt is that gradient descent is an iterative method for finding minima/maxima of a cost landscape. And it uses steepest ascent/descent method. ...
0
votes
1answer
24 views

Is it possible to use the convolution theorem on a finite interval integral ? (Laplace)

Say I have the following equation : $$\int_{0}^{1}\cos(t-\tau)x(\tau) d\tau = t\cos(t)$$ if we replace 1 in the integral for t it is easily solvable using the convolution of Laplace and the answer ...
1
vote
4answers
157 views

How to show $ \lim_{(x,y) \to (0,0)} \frac{3x^2y^2}{x^4+y^4}=\frac{3}{2} $ using the $\epsilon$-$\delta$ notation.

I need to prove that: $$ \lim_{(x,y) \to (0,0)} \frac{3x^2y^2}{x^4+y^4}=\frac{3}{2} $$ using the $\epsilon$-$\delta$ notation. I have tried everything I could think of to make the expression into a ...
0
votes
1answer
19 views

comparing two means using independent samples

A survey will ask baseball fans if they think the Kansas City Royals will return to the World Series this year. We will estimate the difference in the proportion of males and females, ($p_{\texttt{...
7
votes
1answer
1k views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and $...
0
votes
0answers
16 views

How can I use this initial condition for the heat equation

How can I use the following initial condition for a partial differential equation describing heat diffusion? $$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0....
0
votes
2answers
24 views

basic Quantile proves

Let this be my definition of a quantile funktion. X is a real-valued random variable. And let F be it's distribution function. then \begin{align*} F^{-1}(a):=\inf\{x\in \mathbb{R}: F(x) \ge a\}. \end{...
-1
votes
2answers
17 views

Does $1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$ hold?

I am wondering does $1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$? Even if $X_1$ and $X_2$ are dependent?
2
votes
1answer
75 views

Does $\mathrm A \mathrm A^T \succeq x^2 \mathrm I$ imply $\frac{\mathrm A + \mathrm A^T}{2} \succeq x \mathrm I$?

Let $A $ be an $n \times n $ matrix such that $AA^T \geq x^2I, x\geq 0 $, which means that the matrix $AA^T-x^2I$ is positive semidefinite. Can we show that $(A+A^T)/2 \geq xI$? Thanks
1
vote
1answer
18 views

Why is inradius $\times$ surface area equal to thrice the volume?

"Inradius" means radius of largest sphere that is tangent to all faces. For example: Cube - Surface area $= 6a^2$, Inradius $= a/2$, Volume $= a^3$. Sphere - Surface area $= 4\pi r^2$, Inradius $= ...

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