Questions tagged [inverse-problem]

In science an inverse problem is the problem of calculating from a set of observations the causes that produced the observations. Examples are tomography and seismic reconstruction, and many others. Use this tag for statistical methods used with inverse problems.

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inverse of canonical correlation

The canonical correlation obtains the most correlative pairs of components across two matrices. However, the inverse is not commonly known but could be useful for obtaining the most different ...
7 votes
2 answers
856 views

In GD-optimisation, if the gradient of the error function is w.r.t to the weights, isn't the target value dropped since it's a lone constant?

Suppose we have the absolute difference as an error function: $\mathit{loss}(w) = |m_x(w) - t|$ where $m_x$ is simply some model with input $x$ and weight setting $w$, and $t$ is the target value. In ...
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1 answer
28 views

How to decompose a random walk (array) into its Markov Chain transition matrix?

The algorithm, PageRank, receives a Markov Chain transition matrix (page links from one to another.) Either by random walk, or more efficiently, eigenvectors, the stationary distribution of the Markov ...
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22 views

Does blind source separation (ICA) work if channels of mixture are observed asynchronously?

Does Independent Component Analysis (ICA - fastICA, SOBI, etc.) work reliably when applied to a multidimensional mixture (observation) $X = (X^1, \cdots, X^d)$ if the different channels $X^i$ of the ...
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27 views

Mean and covariance estimators for indirect (transformed) observations of Gaussian

Suppose we have a multivariate Gaussian random variable $\mathbf{x}\sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and $n$ measurements $\mathbf{y}_1 = H_1 \mathbf{x}_1, \mathbf{y}_2 = H_2 \mathbf{x}...
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11 views

Optimal threshold decay for L1 homotopy basis pursuit

To solve an underdetermined system of equations where we believe the optimal solution should be the one with minimal L1 norm, we solve the basis pursuit problem: $\min_{\mathbf{x}}{|\mathbf{x}|_1}$ s....
1 vote
1 answer
26 views

Estimate the Image Using Multi Many Realizations of Its Convolution with a Known Filters Using Wiener Filter

Suppose we have a corrupted image $Y = H*X + \epsilon$ that is formed by taking an image $X$, convolving it with a point-spread function $H$, and adding gaussian noise $\epsilon$. Then we know that ...
2 votes
1 answer
47 views

How to invert/infer a parameter in nonlinear conditional expectation function

I wouldn't be surprised if this question has already been asked, as it sounds like a standard bookwork result. However, I'm not sure I know the language to describe it, and when I type in the the ...
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60 views

Inverse Neural Networks

Suppose there is a series of transformation applied to the random variable $z_0$ such that $$ z_M = f_{\theta_{M}} \circ f_{\theta_{M-1}} \circ \ldots \circ f_{\theta_{1}}(z_0) =: f_{\theta}(z_0). $$ ...
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1 answer
84 views

Validation of inverse problem solution based on Bayesian method

Recently, I read a paper about the inverse problem and parameter estimation. The main approach of the paper is based on the Bayesian method. The answer in this method is a posterior probability ...
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87 views

Is it possible to recover original normal distributions from observations of sums of normally distributed variables?

I have been trying to solve the following problem for several weeks. I would greatly appreciate it if you help me solve the following problem. Problem description Assume that there are seven iid ...
1 vote
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165 views

Accuracy of exponential decay parameter estimates comparing different models

I am wondering what approach is more accurate for estimating parameters between two different models of exponentially decaying signal data. The signal decays very rapidly and only a few samples can ...
2 votes
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63 views

Inverse problem with normal distributions and max

Consider $n$ independent normally distributed random variables $$X_i\sim N(\mu_i,\sigma_i^2)$$ and denote $Y = \max\limits_{1\leq i\leq n}\{X_i\}$. We can define the probabilities, for each $1\leq i\...
4 votes
1 answer
117 views

Inferring a Markov chain from its invariant measure

Given a probability measure $p$ on $\{1,\dots,n\}$ assumed to be the invariant measure of some irreducible ergodic Markov chain with unknown transition matrix $P$, i.e., $p = pP$, what (if any) ...
1 vote
2 answers
104 views

Distribution of population size $n$ given binomial sampled quantity $k$ and selection probability $\pi$

Given a drawn (without replacement) sample size $k$ from a binomial distribution with known probability parameter $\pi$, is there a function which gives distribution of likely population size $n$ from ...
1 vote
1 answer
111 views

Existence of least squares and maximum likelihood estimators?

In statistical parameter estimation where there is a deterministic and stochastic component to the observation-generating model, do least squares and maximum likelihood estimators always exist? ...
2 votes
1 answer
1k views

L-curve method for regularization parameter selection

I work on PDE inverse problems and I'm interested in how these can be viewed as problems of statistical inference. I'm looking for some model parameters $m$ which minimize the misfit with some data $d$...
3 votes
0 answers
127 views

Neural Network Inversion and its consequences

I am currently looking at Federated Learning. Here is a good example from google. The idea is that training happens on multiple devices. This means on one hand that training data never leaves a user (...
1 vote
1 answer
1k views

Least square regression with L1 regularization and non-negativity constraint

There are two functions associated by the model $a(x) = \int_{k_1}^{k_2} b(k)\exp(-kx)dk$ where $a(x)$ is the experimental data I have, and $b(k)$ is the information I want to get. Or I can write ...
1 vote
0 answers
673 views

Is the covariance matrix a diagonal matrix with variances on the diagonals?

I am a geophysicist learning about geophysical inverse problems. In many papers, the authors discuss the "covariance matrix" as it applies to the inverse problem. In most geophysical applications, ...
1 vote
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44 views

What is the inference behind the momentum variable and the Kinetic energy for a weakly non-linear inverse problems in the HMC method?

We generate an auxiliary momentum variable in the HMC method to provide gradient for the propagation of trajectory (m, p) (model or position, momentum) in the phase space. If we look into Newton's ...
1 vote
1 answer
68 views

Probability of Crossing a Threshold as a Function of Time (Forward Model)

I know the state of some object at a given time. Let's assume that the state is temperature. At each time, I also know the mean and variance of that temperature. I would like to obtain a probability ...
2 votes
0 answers
20 views

Can we get the input from a multilayer perceptron based on the output of one of its hidden layers?

I was reading a relatively new paper that proposed to split a nerual networks layers into groups and sending each group to different nodes to train them in a distributed manner. In order to not send ...
1 vote
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80 views

Population Monte Carlo Algorithm using L2 Distance Measure/ Likelihood Distribution

I am currently struggling with some concepts of the Population Monte Carlo Framework. Initially, I came across this set of algorithms as I am currently trying to infer parameters from a 7D ...
1 vote
0 answers
77 views

How to properly solve for the inverse problem of OLS? [duplicate]

In textbook ordinary least squares we want to find a vector of coefficients $b_{k+1\times n}$ such that the sum of the squared deviations of what's observed ($y_{n\times 1}$) from what's assumed to be ...
1 vote
1 answer
105 views

What is the error of my regression? [closed]

I'm conducting a polynomial of a third degree upon a diode measurement where Amplification was measured against Voltage. It's a very exponential behavior. However, I used the ...
0 votes
0 answers
34 views

Approximation of fractional function that has real-power numerator

I have the function $f(x)=\frac{(1+x)^k}{1+ax}$, where $x>0, 0<a<k<1$. The function has only one maximum at $x_0=\frac{a-k}{a(k-1)}$, increases on the left of $x_0$ and decreases on the ...
2 votes
0 answers
114 views

Machine learning/Deep learning to solve inverse tomographic problem

The typical simplifiled representation of a tomographic system is $y = Ax$, where $y$ is the collected data (sinogram in CT), $A$ is the projection matrix, and $x$ is the unknown image. The ...
1 vote
0 answers
357 views

How to select the regularization parameter between two losses?

In deep learning, the total loss commonly consists of a task-specific loss and a weight regularized loss: loss = loss_specific + lambda * reg_loss In my case (...
1 vote
0 answers
331 views

How to compare posterior distributions for different observed data? KL-divergence?

So I'm solving an inverse problem with the Bayesian approach $p(u | y) \propto p(y| u )p(u)$. Assuming I have two datasets $y_1$ and $y_2$, what can be said about the difference in the posteriors $p(...
3 votes
1 answer
382 views

Regression with an unknown dependent variable

I want to know if there is any literature about the following regression problem: $$ Y=X\beta +\epsilon$$ where $Y$ is unknown. But, i know $X$ and the OLS estimator of $\beta$ $$ \hat{\beta}=(X^\top ...
2 votes
0 answers
243 views

Connection between MCMC and Optimization for Inverse/Parameter-Estmation Problems

I've been considering two approaches to solving inverse/parameter-estimation problems, and I'm curious to the connection and/or difference between the two approaches. Set up: Say we have a forward ...
1 vote
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348 views

Solving an inverse problem with machine learning

I am running up against a very tough inverse problem that I suspect might be solvable using machine learning. Here is the problem. Overview I am studying an object $X$ which, internally, is ...
0 votes
1 answer
41 views

Marginal Posterior Likelihood-Solving inverse Problem

For a university project, we were required to code our own Parallel Tempering Algorithm and use it to solve an Inverse Problem with 4 Parameters. Unfortunately, I'm not sure if I'm too stupid or have ...
2 votes
0 answers
132 views

Confusion related to inverse problems in statistics [closed]

I am getting started with inverse problems in statistics. However, I didn't something related to it. I was reading this paper http://math.uni-heidelberg.de/studinfo/reiss/CavalierInvProb.pdf. It ...