Questions tagged [linear-algebra]

A field of mathematics concerned with the study of finite dimensional vector spaces, including matrices and their manipulation, which are important in statistics.

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350 views

Finding linear transformation under which distance matrices are similar

I have $n$ sets of vectors, where each set $S_i$ contains $k$ vectors in $\mathbb{R}^d$. I know there is some unknown linear transformation $W$ under which the distance matrix $D_i$ (a $k\times k$ ...
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25 views

Intuition behind understanding eigenvectors and Machine Learning

I am struggling to understand linear algebra application in machine learning, and I am not able to answer the following question. Is the model learned in Machine Learning the eigenvector of the ...
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1answer
28 views

Can all known ML algorithms be written as a sequence of matrix operations?

I keep hearing that machine learning is just linear algebra. Does that mean all known (and all possible?) ML algos, from random forest, to support-vector machines, to recursive neural networks, can ...
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1answer
18 views

What are some theoretical approches to study Machine Learning/Deep Learning?

I've recently studied "basics of machine learning" (chapter 5) in Deep Learning by Ian Goodfellow. They're explaining ML with most of statistical and probabilitical methods. But I wish to ...
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6 views

Projecting 3D Chemical Data Onto a 2D Plane in Motion

I'm trying to model the rotation of two hydrogen atoms about a carbon atom. Say I have a conceptual wheel on an axle that is attached to my car. The axle is described by two points in 3D space, as ...
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1answer
25 views

Why transpose of independent feature matrix is necessary in case of linear regression?

I can follow classical linear regression steps: $Xw=y$ $X^{-1}Xw=X^{-1}y$ $Iw=X^{-1}y$ $w=X^{-1}y$ However, on implementing in Python, I see that instead of simply using ...
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1answer
315 views

Gradient descent formula implementation in python

So I recently started with Andrew Ng's ML Course and this is the formula that Andrew lays out for calculating gradient descent on a linear model. $$ \theta_j = \theta_j - \alpha \frac{1}{m} \sum_{i=1}...
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39 views

How to calculate latent vector for a user in ALS based on some new input?

So I have an ALS trained in pyspark but then I get some interactions from a new user that wasn't in the training set. I want to give recommendations to that new user without retraining the ALS based ...
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4answers
112 views

How to solve Ax = b for A [closed]

Given two know vector x, and b (dimension 3*1 for example), what are the ways to approximate the matrix ...
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2answers
1k views

What is the use of additional column of 1s in normal equation?

Currently I am going through Normal Equation in Machine Learning. $$ \hat\theta = (X^T \cdot X)^{-1} \cdot X^T \cdot y $$ But when I see how they use this equation, I found they always add an ...
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1answer
26 views

Backpropagation with a different sized training set?

I'm trying to create a NN whose input is a (length m) array of 3d vectors $$\vec{x}_i = [x_{i,1},x_{i,2},x_{i,3}], \hspace{5mm}i=1:m $$ and whose output is a similarly sized array: $$\vec{h}_{\theta,...
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What's wrong with my backpropagation through time (BTT) calculation or how to multiple a scaled vector and a matrix without matching dimensions?

I am trying to make a pretty simple RNN from scracth, using only Numpy library of Python. At this moment I am having troubles with BTT as I do not know how to proceed with situation when a ...
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38 views

Workng of LME model used for a set of category variable(s) and a continuous variable?

LME models are being used to analyze the effect of continuos data and category data. Is this model appropriate for checking the effect of two independent variables - one with continuous values and ...
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1answer
43 views

Can I use regression to solve a multiple equation problem

I'm working on a problem which is a multiple equation. I have a group of people and each person in the group is working on different tasks (e.g. n tasks in total). Each person in this group is working ...
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45 views

How to incorporate the uncertainty of the model coefficients in the prediction interval of a multiple linear regression

I'm dealing with the modeling of small experimental data sets. As most experimental work does not generate thousands of samples, but rather a handful, I need to be inventive in how to deal with this ...
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12 views

How effective is Moore Penrose for solving regression problems with overdetermined system of equations?

For regression problems with #Predictors > #observations, I recently read about Moore Penrose (pseudo inverse method) which solves the problem of non invertible matrix in OLS for regression problems. ...
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20 views

Possible flaw in the MDS method for dimensionality reduction

The MDS (multidimensional scaling) method is used to solve the problem of dimensionality reduction. Basically, it does the following: given $n$ points $x_1,\cdots,x_n\in\mathbb R^d$, try to find a ...
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1answer
196 views

How do we define a linearly separable problem?

When we talk about Perceptrons, we say that they are limited for approximating functions that are linearly separable, while Neural Networks that use non-linear transformations are not. I am having ...
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1answer
23 views

optimizing a linear optimization function with linear constarints and binary variables

I am new to optimizations and trying to solve a problem, which I feel falls in the umbrella of optimization. I have an ojective function that needs to be maximized ...
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1answer
33 views

Need explanation of a matrix multiplication

I'm reading the Deep Learning book by MIT. On the page 172, there's a part like this: $$ f^{(1)}(x)=h=W^Tx \tag{1} $$ $$ f^{(2)}(h)=h^Tw \tag{2} $$ Substitute (1) into (2), they got: $$ f(x)=w^TW^Tx $...
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1answer
89 views

Why in this case are gradient steps not perpendicular to contour lines?

There is a theorem that gradient at point is perpendicular to tangent line to contour line at given point. Why in this picture it seems that this rule is not respected? source: http://www....
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34 views

How do I get confidence intervals for an ElasticNet in sklearn?

I need to produce a row for the confidence interval for every field that I am calculating coefficients and scores off of. So here is my code so far- ...
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15 views

What does sparsely compute mean?

I heard someone say a neural network needs to sparsely compute the output. I get what compute means, I get what a sparse matrix is, but what does sparsely compute mean?
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1answer
93 views

PCA formulation - Deep Learning book by Ian Goodfellow

I am reading this deep learning book by Ian goodfellow. In the PCA formulation in the first chapter i.e Linear Algebra, he mentions the following: we need to choose the encoding matrix D. To do so,...
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1answer
44 views

Eigen Decomposition of Data Matrix for PCA

In PCA we Eigen decompose the covariance matrix, not data matrix, Is it because most data matrices are non-square. If they were, isn't is correct to eigen decompose data matrix than the covariance ...
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22 views

Chaining bias terms in backprob

let's say I have a few linear layers $l_1 \dots l_n$: $y=I(\dots I(IX + b_1) + b_2) \dots +b_n)$ where $n$ is sufficiently large and $I$ is the (nonparametric) identity matrix. The gradient for $b_n$...
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1answer
33 views

I can't understand polynomial in the book

I'm reading a book called Bishop - Pattern Recognition and Machine learning. I came across the following equation, in which I don't understand what $W$ stands for. So, what is $W$?
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1answer
430 views

Mathematical formulation of Support Vector Machines?

I'm trying to learn maths behind SVM (hard margin) but due to different forms of mathematical formulations I'm bit confused. Assume we have two sets of points $\text{(i.e. positives, negatives)}$ one ...
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1answer
318 views

Why does np.linalg.eig produce an opposite-signed eigenvector?

I am learning SVD by following this MIT course. In this video, the lecturer is finding the SVD for $$ \begin{pmatrix} 5 & 5 \\ -1 & 7 \end{pmatrix}, $$ which involves finding the ...
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61 views

Linear algebra library for c++

I have been trying to find the substitute of numpy and perform some linear algebra using c++ and here's a list of libraries I have encountered: Eigen Armadillo Dlib GNU Scientific library Please ...
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1answer
299 views

Linear regression with white Gaussian noise

I am new to machine learning , so this question may sound fundamental. My task is to estimate the parameter vector of the equation with the least squares method: $y = \theta_0 + \theta_1x + \theta_2x^...
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19 views

How to add extra constraints to an equation?

Background: I have an equation which looks like as follows: $W \times P = R$ $\left[\begin{array} &{1}&{0}&{0}&-\frac{w_{1}}{w_{o1}} &\dots &{0} &-\frac{w_{1}}{w_{0} } \\...
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1answer
138 views

RNN: why Wx + Uh instead of W[x,h]

Traditionally, a state for RNN is computed as $$h_t = \sigma(W\cdot \vec x + U\cdot \vec h_{t-1} + \vec b)$$ For a RNN, why to add-up the terms $(Wx + Uh_{t-1})$ instead of just having a single ...
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1answer
3k views

How to “reshape” into square matrix for numpy.linalg.solve()?

I'm trying to find the intersection of lines $y=a_1x+b_1$ and $y=a_2x+b_2$ using numpy.linalg.solve(). What I can't get my head around is how to correctly make $A$ ...
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168 views

Connection between piecewise linear basis functions and RELU activation function

ReLU activation is defined as follows $$\sigma(x)=\max(0, x).$$ Let's assume that I have deep network of 1 hidden layer, than output from my layer has form $$ f(x)= \sigma(Wx +b), $$ where matrix W ...
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1answer
58 views

Optimizing vector values for maximum correlation

I'm new to ML, linear algebra, statistics, etc. so bear with me on the terminology... I’m looking to find a vector that produces the maximum correlation for the relationship between 1) all ...
3
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1answer
130 views

How can positional encodings including a sine operation be linearly transformable for any offset?

In the paper "Attention is all you need" the authors add a positional encoding to each token in the sequence (section 3.5). The following encoding is chosen: $ PE(pos, 2dim) = sin(pos / 10000 ^ {2dim/...
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1answer
258 views

What exactly is the “hyperbolic” tanh function used in the context of activation functions?

I know the plot of $\tanh$ activation function looks like. I also know that its output has a range of $[-1, 1]$. Furthermore, I also know the it is defined as follows $$ \tanh(x) = \frac{\sinh(x)}{...
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0answers
74 views

Can we think of neurons as maps between matrices?

Usually when we think about neurons, we imagine that they enact some kind of map between real numbers. For example, a neuron might take in real numbers $x_{i}$ and weight them with parameters $W_{ij}$,...
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0answers
13 views

Structures for incorporating linear functions into a nonlinear optimization problem

I'm working on a problem which naturally involves both linear and nonlinear operations, and I'd like some help understanding the best way to combine these into a neural network framework. To be more ...
2
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1answer
354 views

Machine learning PhD Interview technical questions [closed]

I'm Software Engineer who applied to grad school for Machine Learning/Computer Vision PhD and currently waiting for interview calls. I'm brushing up Linear algebra/ ML topics. What kind of technical ...
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2answers
512 views

How can I implement tangent distance for k-nearest neighbor in python/scikit-learn?

My ultimate aim is to have a function which I can feed into scikit-learn's NearestNeighbor class as a custom metric parameter. ...
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4answers
282 views

Statistics Before Linear Algebra?

I know this is an opinion-based question and will be closed but this is the only place I know that can answer it reasonably and it is a very important matter to me. I am pursuing a machine ...
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2answers
51 views

Derivates with respect to a vector

Suppose I have an equation, $f = X^TY + \dots$ (a few more terms), where $X$ is a vector and $Y$ is a matrix of appropriate dimensions, I want to know how can we take the derivative of $f \text{ w.r.t....
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1answer
335 views

Is there a quick way to speed up ICP in python using a cached KD-tree

I am currently using ICP to match 2 point clouds. These point clouds evolve in time, so I have to repeat this process many times. I am using a standard KD tree from scipy for my nearest neighbor ...
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1answer
42 views

On minimizing matrix norm (AB-C)

Given A, B and C are matrices with dim(A) = m x n, dim(B) = n x p and dim (C) = m x p, the problem asks to evaluate I need to learn $$\tilde{A}$$ such that $$\min_{\tilde{A}}||\tilde{A}^TB-C||$$ and ...
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1answer
54 views

Least Squares Regression $Ax=b$ when $A$ is fixed and $b$ is varied

The typical setting for least squares regression (or over-determined linear system) for $Ax=b$ is to solve $x$ given $A$ and $b$. In other words, $A$ and $b$ are fixed when we solve the problem. My ...
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2answers
32 views

Are euclidian vectors and unit vectors same thing? [closed]

Consider this statement : Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R3. Consider the vectors e1 = (1,0,0), e2 = (0,1,0) and e3 = (0,0,1). Then any ...
5
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1answer
391 views

Closed form solution of linear regression via least squares using matrix derivatives

How is the closed form solution to linear regression derived using matrix derivatives as opposed to using the trace method as Andrew Ng does in his Machine learning lectures. Specifically, I am ...