# derivative of norm 1

Can you introduce me a source to master this material. $$,$$ The possibility to use a vectorized operation is a huge advantage over the other norms. First, let's calculate the dot product of the vectors: This is the same results as with the dot product. The submultiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality. A perfect model would have only 0's while a very bad model would have huge positive or negative values. Remember (and try it to convince yourself) that the order the vector in the dot product matters: It should be our squared Euclidean norm! So let me plug in 9, we have 1/2, 9 to the -1/2. If I understand correctly, you are asking the derivative of $\frac{1}{2}\|x\|_2^2$ in the case where $x$ is a vector. 2 Let A;B be the derivative at x \norm{\bs{x}}_0=(\sum_i|\bs{x}_i|^0)^{1/0} \norm{\bs{x}}_\infty = \max\limits_i|x_i| We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, … \newcommand\bs[1]{\boldsymbol{#1}} a Banach space.Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for … \cdots\\ \end{bmatrix} The goal of this tutorial is to enter mathematics for data science by coding with Python/Numpy. Let's have the following vector: We have seen that its squared $L^2$ norm is calculated with: Then, to calculate the partial derivatives, we consider all other variables as constant. We have seen that norms are nothing more than an array reduced to a scalar. 1 & 6 $$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 = \sum_{i=1}^{n} \lvert x_i - (A\mathbf x)_i\rvert = \sum_{i=1}^{n} \lvert x_i - A_i \cdot \mathbf x \rvert = \sum_{i=1}^{n} \lvert x_i - \sum_{j=1}^n a_{ij} x_j\rvert$$ Apart from a sign error, your result looks correct. $$,$$ Therefore this norm corresponds to the number of non-zero elements in the vector. So x² plus 4x, it's the low part. The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. So I’m looking for the derivative because, remember, the critical points are points where the derivative equals 0 or is undefined. $$,$$ For instance, the partial derivative according to $u_1$ is the derivative of $u_1^2+a$ ($a$ being the constant corresponding to all other variables). frobenius norm derivative, The Frobenius norm is an extension of the Euclidean norm to {\displaystyle K^ {n\times n}} and comes from the Frobenius inner product on the space of all matrices. This will just convert the initial column vector to a row vector: We can calculate the transpose of $\bs{x}$ with the T method of Numpy objects: The dot product of $\bs{x}$ and $\bs{x}^\text{T}$ (see here if you need some reminder about the dot product) corresponds actually to the multiplication of each element by itself: This is exactly the definition of the squared Euclidean norm! Writing explicitly the dependence of y on x, and the point at which the differentiation takes place, the formula for the derivative of the inverse becomes (in Lagrange's notation): [−] ′ = ′ (− ()). The error vectors are multidimensional: there is one dimension per observation. \end{bmatrix} One way to calculate the length of the vectors is to use the Pythagorean theorem: $\sqrt{x^2+y^2}$. u_2\\ So the first thing we want to do is recall the definition of the derivative function. Yes it does, but we will not prove this fact. \dfrac{d\norm{\bs{u}}_2}{du_2} = \dfrac{u_2}{\sqrt{(u_1^2+u_2^2+\cdots+u_n^2)}}\\ See below. If $$f(x)$$ is both invertible and differentiable, it seems … \cdots \\ Every (real or complex) vector space admits a norm: If x• = (xi)i ∈ I is a Hamel basis for a vector space X then the real-valued map that sends x = ∑i ∈ I sixi ∈ X (where all but finitely many of the scalars si are 0) to ∑i ∈ I |si| is a norm on X. Note 2: we used the colors from seaborn manually with sns.color_palette(). 0\times2+2\times2 = 4 We have seen above that one condition for a function to be a norm is that it respects the triangle inequity. There are no particular prerequisites, but if you are not sure what a matrix is or how to do the dot product, the first posts (1 to 4) of my series on the deep learning book by Ian Goodfellow are a good start. 3 \\ -1 x to the, then I subtract 1 from the exponent -1, minus another 1 is -2. Choosing which norm to use depends a lot of the problem to be solved since there are some pros and cons for applying one or another. This may be the more used norm with the squared $L^2$ norm (see below). A major result that uses the L p,w-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals. The Derivative of an Inverse Function. In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. considering that the derivative of x with respect to x is 1. \norm{\bs{u}}_2^2 = (\sqrt{\sum_i \bs{x}_i^2})^2 = \sum_i\bs{x}_i^2 The $z$-axis corresponds to the value of the norm and the $x$- and $y$-axis corresponds to two parameters. Here, $k$ is a scalar and $\bs{u}$ a vector. It is what we had used when we calculated the length of our vectors with the Pythagorean theorem above. Any hints are appreciated, thanks! \norm{\bs{u}+\bs{v}} \leq \norm{\bs{u}}+\norm{\bs{v}} So plus x to the -2. The model with the smaller total error is, the better: It looks like the model 1 is far better than the model 2. It is the $L^\infty$ norm and corresponds to the absolute value of the greatest element of the vector. So the derivative is going to be 1/2x to the -1/2. If the derivative is a higher order tensor it will be computed but it cannot be displayed in matrix notation. Consider \ ( (2^x-1)/x\) for some small values of \ (x\): 1, \ (0.828427124\), \ (0.756828460\), \ (0.724061864\), \ (0.70838051\), \ (0.70070877\) when \ (x\) is 1, \ (1/2\), \ (1/4\), \ (1/8\), \ (1/16\), \ (1/32\), respectively. \norm{\bs{x}}_p=(\sum_i|\bs{x}_i|^p)^{1/p} $$Let's start with a vector \bs{x}: As usual, we will use code to check the process. We can calculate the length of the two vectors: Congratulation!$$ =-a_{1k}sign(x_1 - \sum_{j=1}^n a_{1j} x_j)-\cdots+(1-a_{kk})sign(x_k - \sum_{j=1}^n a_{kj} x_j)-\cdots -a_{nk}sign(x_n - \sum_{j=1}^n a_{nj} x_j)$$. Here are the operations using numpy. We will see in this example that the squared Euclidean norm can be calculated with vectorized operations. This is equivalent to take the L^2 norm of the matrix after flattening. We have also noticed that there are some variations according to the function we can use to calculate it. \bs{u}= The exponent comes out in front. Imagine that you have a dataset of songs containing different features. There can be performance gain due to optimization. Hence, lim jjhjj!0 jhTAhj jjhjj lim jjhjj!0 jjhjjjjAjj 2jjhjj jjhjj lim jjhjj!0 jjAjj 2jjhjj= 0 2. 3 plus 1 … 0 \\$$, $$In the case of the L^2 norm, the derivative is more complicated and takes every element of the vector into account. You can think of the norm as the length of the vector. Our vectors are: We will see later different types of the norm, but for now on, we will use a classic one: the Euclidean Norm (L^2). The L^2 norm (or the Frobenius norm in case of a matrix) and the squared L^2 norm are widely used in machine learning, deep learning and data science in general.$$. The cost function is a function that represents the error of your model, so you want this error to be as small as possible. By examining the TV minimization with Euler-Lagrange equation, e.g,, Eq. (max 2 MiB). For this reason, we have the following partial derivatives: What is great about the gradients of the squared $L^2$ norm is that the derivatives do not depend on the other variables. $$,$$ 2 \\ 2 is the spectral norm of A. Norms respect the triangle inequality. is comparable to the L p,w-norm. H110 NORMlite November 14, 1999 5:50 pm Prof. W. Kahan Page 1/21 In this tutorial, we will approach an important concept for machine learning and deep learning: the norm. In this case, the vector is in a 2-dimensional space, but this also stands for more dimensions. $$,$$ CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A+ The pseudo inverse matrix of the matrix A (see Sec. Let's start by calculating the norm with the formula: By the way, remind that the $L^2$ norm can be calculated with the linalg.norm() function from Numpy: Here is the graphical representation of the vector: We can see that the vector goes from the origin (0, 0) to (3, 4) and that its length is 5. The absolute value is used because a negative error (true duration smaller than predicted duration) is also an error. We begin by considering a function and its inverse. D high is the derivative of the upper part. 1 over 3 times 1/2 is 1/6. The length of the error vector of the first model is $22.36$ and the length of the error vector of the second model is around $16.64$. We can note that the absolute value is not needed anymore since $x$ is squared. We can see that there is a sum of elements so we can think of it as an iteration over the $i$ elements: This will be clear with examples using these widely used $p$-norms. I correct the sign error. The norm will map the vector containing all your errors to a simple scalar, and the cost function is this scalar for a set of value for your parameters. \norm{\bs{u}+\bs{v}} = \sqrt{(1+4)^2+(6+2)^2} = \sqrt{89} \approx 9.43 3 \\ Further in the case p > 1, this expression defines a norm if r = 1. I want to compute the following derivative with respect to $n\times1$ vector $\mathbf x$. \end{bmatrix} For example, let f(x)=7x 3-8x 2 +2+4e x.By using the power rule, the derivative of 7x 3 is 3*7x 2 =21x 2, the derivative of -8x 2 is 2*(-8)x=-16x, and the derivative of 2 is 0. These \things" include taking derivatives of multiple components simultaneously, taking derivatives in the presence of summation notation, and applying the chain rule. \begin{cases} \end{bmatrix} I am going through a video tutorial and the presenter is going through a problem that first requires to take a derivative of a matrix norm. $$,$$ The Derivative of an Inverse Function. \norm{\bs{y}}_2=\sqrt{2^2+2^2}=\sqrt{8} The $L^2$ norm is: Let's calculate the derivative of it according to $u_1$: We can see that the partial derivative of $u_1$ still contains $u_2...u_n$. You have the following results in seconds for 7 observations: These differences can be thought of as the error of the model. $$,$$ This means that the $L^2$ norm is more sensible to outliers since significant error values will give enormous squared error values. 1-norm norm2() Euclidean norm. This is why this is crucial to be able to calculate the derivative efficiently. It is not really a norm because if you multiply the vector by $\alpha$, this number is the same (rule 4 above). Let's start with the way we would use it. 2 & 5 & 3 & 3 Using the power $0$ with absolute values will get you a $1$ for every non-$0$ values and a $0$ for $0$. Just note that we use the function deg2rad from Numpy because np.cos takes the angle in radian, so we have to do the conversion. In this post, we explore several derivatives of logarithmic functions and also prove some commonly used derivatives. We have first calculated the sum of the errors, but we can also use the Pythagorean theorem to compute the norm of a vector. $$,$$ Norms are $0$ if and only if the vector is a zero vector. It just means that the norm of the sum of some vectors is less than or equal to the sum of the norms of these vectors: To show what this means, we will take two vectors, each containing two elements (again, useful to be represented as x and y coordinates). @PeterK., user153245: That question came out of interest about the background of the original question; I'm very well aware the needs to find a derivate of some norm, metric etc, but usually, when questions like OP's are asked, there's a whole interesting problem to solve behind that :). \begin{bmatrix} \norm{\bs{x}}_2=\sqrt{0^2+2^2}=\sqrt{4}=2 It is still quite hard to represent 7 dimensions so let's again simplify the example and keep only 2 observations: Now we can represent these vectors considering that the first element of the array is the x-coordinate and the second element is the y-coordinate. The last thing to setup is the appearance of the graphics that we will create: Let's start with a simple example. \begin{bmatrix} \dfrac{d\norm{\bs{u}}_2^2}{du_2} = 2u_2\\ \norm{\bs{u}}_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2} The better model is just the model corresponding to the smaller vector. \end{bmatrix} = Math. 3 Having some comprehension of these concepts can increase your understanding of various algorithms. 2 \\ We're talking about the derivative function. A natural way would be to take the sum of the absolute values of these errors. Different functions can be used, and we will see a few examples. See here and here for more details. The squared $L^2$ norm is convenient because it removes the square root and we end up with the simple sum of every squared value of the vector. There are other methods to derive (prove) the derivatives of the inverse Trigonmetric functions. u_n You have just calculated the norm of the error vector for each model! $$,$$ Linear algebra is one of the basic mathematical tools that we need in data science. Mathematica is actually capable of computing the Norm of a Quaternion but does someone know whether it can expand the Norm function so that the above input yields the expanded derivative (//FunctionExpand did not change anything on the output...). However, this can be ambiguous in some cases. $$,$$ Now we need the slope, particularly we need the slope at 9. By the end of this tutorial, you will hopefully have a better intuition of this concept and why it is so valuable in machine learning. This means that there are multiple functions that can be used as norms. I need help understanding the derivative of matrix norms. We want a function to help us plot the vectors. Implicit differentiation can also be employed to find the derivatives of logarithmic functions, which are of the form $$y = \log_a{x}$$. And remember the quotient rule; it's low d high. Now we want to go a step further and know how we can change the parameters of our model to reduce the overall error. \begin{bmatrix} So what I have here is 3, -1 times -1 is +1. We can see this by graphically comparing the squared $L^2$ norm with the $L^2$ norm. $$,$$ Remark: Not all submultiplicative norms are induced norms. How I can represent the answer compactly? If only 2 dimensions are specified in the vector, it starts at (0, 0). We will also see how the derivative of the norm is used to train a machine learning algorithm. (2.5a) in [1], you would see the answer. We will plot the vectors $\bs{u}$, $\bs{v}$ and $\bs{u}+\bs{v}$ using our plotVectors function and adding some text to identify the vectors: The length of $\bs{u}$ plus the length of $\bs{v}$ is larger than the length of the vector $\bs{u}+\bs{b}$. Posted by 3 years ago. $$,$$ 5 \\ Let's check with Numpy. The same thing is true with more than 2 dimensions, but it would be hard to visualize it. $$,$$ Norms are useful here because it gives you a way of measuring this error. Now imagine that you try another model and you end up with the following differences between predicted and real song durations: What can you do if you want to find the best model? $$,$$ The norm of a vector multiplied by a scalar is equal to the absolute value of this scalar multiplied by the norm of the vector. Under the hood, we iterate on this array of vectors and use plt.quiver() to plot them. $$,$$ You just have calculated another norm of the error vectors. \end{cases} There are also a large number of norms that exhibit additional properties that make them useful for specific problems. \dfrac{d\norm{\bs{u}}_2}{du_n} = \dfrac{u_n}{\sqrt{(u_1^2+u_2^2+\cdots+u_n^2)}}\\ \begin{bmatrix} \begin{bmatrix} Archived. 4 & 2 $$,$$ Indeed, a big advantage of the squared $L^2$ norm is that its partial derivative is easily computed. These plots have been done with the help of this website. These are two different norms, and it shows that there are multiple ways of calculating the norms. Basic Setup. Theorem 6. F'(x) is the limit as h approaches 0 of f(x+h) minus f(x) over h. To obtain the Gradient of the TV norm, you should refer to the calculus of variations. \end{bmatrix} \cdot \norm{\bs{u}}_2 = \sqrt{(u_1^2+u_2^2+\cdots+u_n^2)} = (u_1^2+u_2^2+\cdots+u_n^2)^{\frac{1}{2}} The dot product between the vectors $\bs{x}$ and $\bs{y}$ can be retrieved with the $L^2$ norms of these vectors. $$= \frac{\partial }{\partial x_k}\bigg(\lvert x_1 - \sum_{j=1}^n a_{1j} x_j\rvert +\cdots+ \lvert x_k - \sum_{j=1}^n a_{kj} x_j\rvert + \cdots\lvert x_n - \sum_{j=1}^n a_{nj} x_j\rvert \bigg)$$ \bs{x^\text{T}y}= and A2 = 2 2 2 2! x= (1;0)T. Example of a norm that is not submultiplicative: jjAjj mav= max i;j jA i;jj This can be seen as any submultiplicative norm satis es jjA2jj jjAjj2: In this case, A= 1 1 1 1! \bs{u}= The other gradients follow the same structure: The squared $L^2$ norm is great, but one problem with it is that it hardly discriminates between 0 and small values because the function increases slowly. An example is the Frobenius norm. The derivative with respect to $x$ of that expression is simply $x$ . $$,$$ $$,$$ The term with $(1-a_{1k})$ should have a positive sign. It is what we have used intuitively at the beginning of this tutorial: The Euclidean norm is the $p$-norm with $p=2$. It really is an excellent basis for a data science/machine learning path. The gradient descent algorithm can be used to find the minimum of this function. Here is a problem. \bs{x}^\text{T}= @Marcus Müller, $L_1$ norm is used as a regularization term in reconstructing signal and image. We have seen that the norms can be used to evaluate the goodness of a model by summarizing the vectors of errors. \dfrac{d\norm{\bs{u}}_2}{du_1} = \dfrac{u_1}{\sqrt{(u_1^2+u_2^2+\cdots+u_n^2)}}\\ \bs{v}= \norm{\bs{x}}_2=(\sum_i \bs{x}_i^2)^{1/2}=\sqrt{\sum_i \bs{x}_i^2} Geometrically, this simply means that the shortest path between two points is a line! 2 Go and plot these norms if you need to move them in order to catch their shape. So we have one error vector for each model. \bs{x}= (Properties) (1) Addition Let f : R n!R mand g : R !R be two differentiable functions. Remember that the derivative of e x is itself, e x.So, by using the sum rule, you can calculate the derivative of a function that involves an exponential term. \end{bmatrix} We will see later in details what is the $L^1$ or $L^2$ norms. $$,$$ $$,$$ One way to find the better line is to start with random parameters and iterate by minimizing the cost function. $$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1$$. 1 Simplify, simplify, simplify Much of the confusion in taking derivatives involving arrays stems from trying to do too many things at once. We will see that it is not the case with the $L^2$ norm. In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L p-norms of the function together with its derivatives up to a given order. \norm{\bs{u}}_2^2 = u_1^2+u_2^2+\cdots+u_n^2 For instance, the $L^1$ norm is more robust than the $L^2$ norm. \bs{x}^\text{T}\bs{y} = \norm{\bs{x}}_2\cdot\norm{\bs{y}}_2\cos\theta where $\mathbf{I}$ is the $n\times n$ identity matrix. Let's say that you want to fit a line to a set of data points. Click here to upload your image \begin{bmatrix} But in the paper I study, there is $A^T$ instead $A$ in the first parenthesis. We will see later the pros and cons of these different norms. We call $p$-norm the following category of functions that depend on $p$: Let's dive into this equation step by step. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Before all, we will setup the libraries that we will use: And add some Latex shortcut to the commands bs for bold symbols and norm for the symbol of the norm:  To have an idea of the graphical representation of this, let's take our preceding example again. u_1\\ If you think of the norms as a length, you can easily see why it can't be negative. 1. \cdots\\ We can formulate an LP problem by adding a vector of optimization parameters which bound derivatives: We have seen the conditions required by the function to be called norm.