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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18 views

How does the equation “dW = - (2 * (X^T ).dot(Y - Y_hat)) / m” comes in Linear Regression (using Matrix + Gradient Descent)?

I was trying to code the Linear Regression in Python using Matrix Multiplication method using Gradient Descent and followed a code where there was no mention what is the loss but just a code as Per ...
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How do I find the eigenvectors corresponding to the largest eigenvalue of a matrix in scikit?

Im trying to determine the principal component 1 and 2 of a symmetric matrix using sklearn. Id appreciate any help. Thank you.
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How does tree-based algorithms handle linearly combined features?

While I am aware that tree-based algorithms (e.g., DT, RF, XGBoost) are 'immune' to multi-collinearity, how do they handle linearly combined features? For example, is there is any additional value or ...
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Geometric classification models

In class we have been presented with a Geometric classification model such that the goal is to construct a linear decision boundary $\bf{w} \cdot \bf{x} = t$; where $\bf{w}$ is the vector from the ...
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63 views

Dot product and linear regression

I'm studying PCA and my professor said something about finding the linear regression by doing the dot product of both axis. Could someone explain to me why? The dot product returns a number. What's ...
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Why is this equation converted to matrix form in this way? Is it possible to multiply an inverse matrix with a vector?

I have been banging my head on wall for days trying to decode this equation. please help me out with this... Below is the equation (consider $x$ as $\Delta x$, and $y$ as $\Delta y$): $x = - \eta(Id-\...
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53 views

Why do we usually have fully connected layers of same sizes in CNNs?

Is there any specific reason that we observe in CNNs, the fully connected layers usually have the same sizes? You can verify this for many CNNs. I'm aware that if, for instance, we have a vector of ...
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49 views

Hypothesis vs Hyperplane in Machine Learning

I am finding it hard to understand the clear difference between Hypothesis and Hyperplane. I know that Hypothesis is a candidate model that maps inputs to outputs after training . And , Hyperplane is ...
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Confusion with Notation in the Book on Deep Learning by Ian Goodfellow et al

In chapter 6.1 on 'Example: Learning XOR', the bottom of page 168 mentions: The activation function $g$ is typically chosen to be a function that is applied element-wise, with $h_i = g(x^TW_{:,i}+c_i)...
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Deriving vectorized form of linear regression

We first have the weights of a D dimensional vector $w$ and a D dimensional predictor vector $x$, which are all indexed by $j$. There are $N$ observations, all D dimensional. $t$ is our targets, i.e, ...
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SVD Kernel and Linear Algebra Kernel, is there a conceptual difference?

Is the term kernel used in Sklearn to execute the SVD machine learning algorithm conceptually related to the notion of a kernel in linear algebra ( null space )? Or do they happen to use this same ...
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115 views

Removing constant from the regression model

I am trying to calibrate two variables $(X,Y)$ of different measuring techniques from two instruments, the result of the linear regression analysis appears as shown in the image. The result shows the ...
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27 views

Understanding the algebra behind a specific partial derivative equation

I am following this article about neural networks. Given: Until here I understand everything, but then he continues to: I don't understand how he got to that conclusion. I think he skipped some ...
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35 views

Human intuition behind SVD in case of recommendation system

I checked the SVD for recommendation engine thread but it does not answer my question. I struggled very hard to understand the SVD from a linear-algebra point of view. But in some cases I failed to ...
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2answers
160 views

Dose finding slope/intercept using the formula of m,b gives best fit line always In linear regression?

In liner regression We have to fit different lines and chose one with minimum error so What is the motive of having a formula for m,b that can give slope and intercept value in the regression line ,...
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Normal equation for linear regression is illogical

Currently I'm taking Andrew Ng's course. He gives a following formula to find solution for linear regression analytically: $θ = (X^T * X)^{-1} * X^T * у$ He doesn't explain it so I searched for it and ...
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When is it useful to measure the Frobenius norm of a matrix?

In Deep Learning section 2.5 the author review some measures for the size of vectors and matrices. When in general is it useful for someone to know these? For instance they give the example of the ...
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Predicting a variably-placed value in a vector

I have $m$ vectors in $\mathbb{R}^n$, where $m >> n$, and I want to train a model to impute a value $x_i$ in $\mathbf{x}$, where $1 \leq i \leq n$ (and can vary by vector). For instance, I may ...
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Intuition behind understanding eigenvectors and Machine Learning [closed]

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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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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47 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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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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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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43 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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2answers
105 views

Linear regression with a fixed intercept and everything is in log

I have a set of values for a surface (in pixels) that becomes bigger over time (exponentially). The surface consists of cells that divide over time. After doing some modelling, I came up with the ...
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2answers
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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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1answer
54 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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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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41 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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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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746 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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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
36 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
189 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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507 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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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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431 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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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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572 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
61 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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1answer
135 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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484 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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1answer
547 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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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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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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64 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 ...
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202 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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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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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 ...