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.
63
questions
0
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0answers
14 views
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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0answers
15 views
In Ax = b, why does x not pass through the origin? [migrated]
I am facing some difficulty in understanding the concept of x being a vector which doesn't pass through the origin.
Let me tell you what I understand :
A subspace is a space that contains all linear ...
0
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1answer
52 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 ...
0
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1answer
24 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 ...
0
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1answer
24 views
Human intuition behind SVD in case of recommendation system
This 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 connect the dots. So, I started to see all the ...
0
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2answers
75 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 ,...
0
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1answer
49 views
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 ...
0
votes
1answer
21 views
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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0answers
10 views
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 ...
1
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0answers
35 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 ...
-1
votes
1answer
21 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 ...
0
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0answers
8 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 ...
0
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0answers
40 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 ...
1
vote
1answer
31 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 ...
2
votes
4answers
123 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 ...
1
vote
1answer
28 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
...
1
vote
2answers
26 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 ...
1
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1answer
30 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,...
1
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0answers
19 views
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 ...
1
vote
0answers
46 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 ...
1
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0answers
14 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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0answers
39 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 ...
3
votes
0answers
21 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 ...
4
votes
2answers
298 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 ...
2
votes
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
...
2
votes
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
$...
2
votes
1answer
111 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....
0
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0answers
42 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-
...
0
votes
1answer
377 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.
$$ \theta_j = \theta_j - \alpha \frac{1}{m} \sum_{i=1}...
0
votes
0answers
16 views
What does sparsely compute mean?
I heard someone say a neural network needs to sparsely compute the output.
I get what compute means, I get what a sparse matrix is, but what does sparsely compute mean?
13
votes
1answer
376 views
Finding linear transformation under which distance matrices are similar
I have $n$ sets of vectors, where each set $S_i$ contains $k$ vectors in $\mathbb{R}^d$. I know there is some unknown linear transformation $W$ under which the distance matrix $D_i$ (a $k\times k$ ...
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votes
1answer
36 views
I can't understand polynomial in the book
I'm reading a book called Bishop - Pattern Recognition and Machine learning.
I came across the following equation, in which I don't understand what $W$ stands for.
So, what is $W$?
4
votes
1answer
466 views
Mathematical formulation of Support Vector Machines?
I'm trying to learn maths behind SVM (hard margin) but due to different forms of mathematical formulations I'm bit confused.
Assume we have two sets of points $\text{(i.e. positives, negatives)}$ one ...
1
vote
1answer
48 views
Eigen Decomposition of Data Matrix for PCA
In PCA we Eigen decompose the covariance matrix, not data matrix, Is it because most data matrices are non-square. If they were, isn't is correct to eigen decompose data matrix than the covariance ...
3
votes
1answer
104 views
PCA formulation - Deep Learning book by Ian Goodfellow
I am reading this deep learning book by Ian goodfellow. In the PCA formulation in the first chapter i.e Linear Algebra, he mentions the following:
we need to choose the encoding matrix D. To do so,...
1
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1answer
351 views
Why does np.linalg.eig produce an opposite-signed eigenvector?
I am learning SVD by following this MIT course. In this video, the lecturer is finding the SVD for
$$
\begin{pmatrix} 5 & 5 \\ -1 & 7 \end{pmatrix},
$$
which involves finding the ...
0
votes
1answer
340 views
Linear regression with white Gaussian noise
I am new to machine learning , so this question may sound fundamental. My task is to estimate the parameter vector of the equation with the least squares method:
$y = \theta_0 + \theta_1x + \theta_2x^...
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0answers
19 views
How to add extra constraints to an equation？
Background：
I have an equation which looks like as follows:
$W \times P = R$
$\left[\begin{array} &{1}&{0}&{0}&-\frac{w_{1}}{w_{o1}} &\dots &{0} &-\frac{w_{1}}{w_{0} } \\...
1
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1answer
4k views
How to “reshape” into square matrix for numpy.linalg.solve()?
I'm trying to find the intersection of lines $y=a_1x+b_1$ and $y=a_2x+b_2$ using numpy.linalg.solve(). What I can't get my head around is how to correctly make $A$ ...
1
vote
0answers
173 views
Connection between piecewise linear basis functions and RELU activation function
ReLU activation is defined as follows
$$\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 ...
1
vote
1answer
59 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
votes
1answer
146 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/...
1
vote
0answers
76 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 $W_{ij}$,...
1
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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
votes
1answer
394 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 ...
1
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2answers
551 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.
...
3
votes
0answers
63 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 ...
1
vote
4answers
347 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 ...
12
votes
3answers
5k views
How does tensor product/multiplication work in TensorFlow?
In Tensorflow, I saw the following example:
...
0
votes
1answer
45 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 ...
