Eigen Binary Download
Pirjo Unzicker
In eigen there are component-wise binary operators, like +,- etc. The are also two class functions cwiseMin and cwiseMax, but I would like to extend it with, at least, AbsDiff xi - yi ans SqrDiff (xi - yi)^2. But how to do it?
The only thing I could think of is to retrieve the two data pointers and operate on them (maybe, using cilk), but looks like a hack to me. I was expecting the possibility to define scalar functions like AbsDiff(x,y)=x-y ans SqrDiff(x,y)=(x-y)^2 and pass them to a generic eigen traversal routine, but apparently there is'nt. am I missing something, maybe simpler?
I currently have a bool mask vector generated in Eigen. I would like to use this binary mask similar as in Python numpy, where depending on the True value, i get a sub-matrix or a sub-vector, where i can further do some calculations on these.
We use the CMake build system, but only to build the documentation and unit-tests, and to automate installation. If you just want to use Eigen, you can use the header files right away. There is no binary library to link to, and no configured header file. Eigen is a pure template library defined in the headers.
There is also a private mailing list which should only be used if you want to write privately to a few core developers (it is read by Gaël, Christoph, Rasmus, Antonio, David, and Constantino). The address is eigen-core-team at the same lists server as for the Eigen mailing list. You do not need to subscribe (actually, subscription is closed). For all Eigen development discussion, use the public mailing list or the issue tracker on GitLab instead.
Consider $A$, a random binary matrix of zeros and ones in $\mathbbR^M\times N$, and $M>N$. We assume that $P(a_i,j=0)=P(a_i,j=1)=0.5$ (although I appreciate any advice on the case of non-even probabilities). Are there any results that provide a lower bound (and maybe an interesting upper bound) on the eigenvalues of $A^TA$? The reference ring is certainly $\mathbbR$.
Since the mean entry will be equal to $0.5$, then the largest eigenvalue will be (approximately) equal to $0.25MN$, and the corresponding eigenvector will be the vector of all $1$'s (again, approximately). The rest of the eigenvalues will follow Marchenko-Pastur. In this case the random variables in the matrix have variance $\sigma^2=0.25$, which means that the next largest eigenvalue will be $0.25(1+\sqrtN/M)^2M$ and the smallest will be$0.25(1-\sqrtN/M)^2M$. In other words, the range of eigenvalues other than the first will be$$\bigl[0.25(\sqrtM-\sqrtN)^2,0.25(\sqrtM+\sqrtN)^2\bigr].$$
I need to store large sparse matrices in Eigen. I cannot find anything in the library except the function below, in Eigen/Unsupported. The problem with saveMarket is, that it saves in text format. Due to the size of my matrices I need to store my sparse matrices as binaries. Is there an easy way to adjust the function below to store as a binary. And an easy way to reload the matrix?
An interesting class of optimization problems to be addressed by quantum computing are Quadratic Unconstrained Binary Optimization (QUBO) problems. Finding the solution to a QUBO is equivalent to finding the ground state of a corresponding Ising Hamiltonian, which is an important problem not only in optimization, but also in quantum chemistry and physics. For this translation, the binary variables taking values in \(\0, 1\\) are replaced by spin variables taking values in\(\-1, +1\\), which allows one to replace the resulting spin variables by Pauli Z matrices, and thus, an Ising Hamiltonian. For more details on this mapping we refer to [1].
In the following we first illustrate the conversion from a QuadraticProgram to a SparsePauliOp and then show how to use the MinimumEigenOptimizer with different MinimumEigensolvers to solve a given QuadraticProgram. The algorithms in Qiskit optimization automatically try to convert a given problem to the supported problem class if possible, for instance, the MinimumEigenOptimizer will automatically translate integer variables to binary variables or add linear equalityconstraints as a quadratic penalty term to the objective. It should be mentioned that a QiskitOptimizationError will be thrown if conversion of a quadratic program with integer variables is attempted.
Then, I tried to correlate the eigengene values of the modules with the external trait information. I was insecure if I can use correlation for binary information (my only "trait" information is if the sample is from human with/ without disease). However, Ive read all posts I was able to find about this and there it was discussed that this is possible and basically a t-test between the groups.
My question of interest was if I could find differentially co-expressed modules between disease/healthy persons. And I have additional data from 2 other time points which I thought to maybe analyze with consensus analysis/ eigengene networks...
PS: I also had a look at CEMiTool, which is somewhat based on WGCNA I think... There they seem to do gene set enrichment analysis for group differences. As far as I understood it mathematically, the difference is that there the expression of the whole module goes in, not only the eigengene. I tried this for my sample and every module was significant... I am really confused.
Additionally, I was not sure if spearman would be the better option, as the diagnostics when I used pearson were not optimal I think. This is the qqplot.However, when I used spearman I had the strange case, that in one module the NES was 0 in both groups, but nevertheless the p-value was significant! What does that mean?? Didnt I get it right that in CEMiTool a significant p-value in GSEA means that the ranked mean expressions from genes in this module is different between the groups? While in WGCNA sth like a PCA is applied first and the first eigengene is compared between groups?
About the group comparisons, as you said before, both packages use very different methods, so different outcomes are to be expected. It would be interesting to see how WGCNA's eigengene approach relates to the GSEA approach though, maybe you could enlighten us (:
I have a family of Cayley graphs of a certain class of groups, and I want to test my hypothesis that they form a family of expanders. To do that, I'll need to compute the second-largest eigenvalue of their adjacency matrices. Unfortunately, these graphs rapidly become absolutely enormous - their size grows super-exponentially. To look at even the second non-trivial case, I'm dealing with an 4M dimensional adjacency matrix, and the third non-trivial case is over 3 billion. Unless I can get time on a supercomputer, the usual O(n3) Schur decomposition methods are going to be out of reach, and I'll only be able to do the first non-trivial case.
Still there's one problem left to solve. Imagine we are given \(400\) images sized \(100 \times 100\) pixel. The Principal Component Analysis solves the covariance matrix \(S = X X^T\), where \(size(X) = 10000 \times 400\) in our example. You would end up with a \(10000 \times 10000\) matrix, roughly \(0.8 GB\). Solving this problem isn't feasible, so we'll need to apply a trick. From your linear algebra lessons you know that a \(M \times N\) matrix with \(M > N\) can only have \(N - 1\) non-zero eigenvalues. So it's possible to take the eigenvalue decomposition \(S = X^T X\) of size \(N \times N\) instead:
The resulting eigenvectors are orthogonal, to get orthonormal eigenvectors they need to be normalized to unit length. I don't want to turn this into a publication, so please look into [73] for the derivation and proof of the equations.
So some research concentrated on extracting local features from images. The idea is to not look at the whole image as a high-dimensional vector, but describe only local features of an object. The features you extract this way will have a low-dimensionality implicitly. A fine idea! But you'll soon observe the image representation we are given doesn't only suffer from illumination variations. Think of things like scale, translation or rotation in images - your local description has to be at least a bit robust against those things. Just like SIFT, the Local Binary Patterns methodology has its roots in 2D texture analysis. The basic idea of Local Binary Patterns is to summarize the local structure in an image by comparing each pixel with its neighborhood. Take a pixel as center and threshold its neighbors against. If the intensity of the center pixel is greater-equal its neighbor, then denote it with 1 and 0 if not. You'll end up with a binary number for each pixel, just like
CheckInstall keeps track of all the files created ormodified by your installation script ("make install""make install_modules", "setup", etc), builds astandard binary package and installs it in yoursystem giving you the ability to uninstall it with yourdistribution's standard package management utilities.
The checkinstall command is calls the make install command. It monitors the files that are installed and creates a binary package from them. It also installs the binary package with the Linux package manager.
Principal component analysis (PCA) is a method commonly used for dimensionality reduction. The eigenvectors of the covariance matrix are called principal components. Heres a thorough tutorial on PCA and applied to computer vision (Lindsay Smith, 2002).
The partial PCA calculated the largest eigenvalue of 2942.0060 and its eigenvector [0.3997, -0.9166]. By a slight difference, image moments returned the largest eigenvalue of 2943.2583 and its eigenvector of [0.4027, -0.9153].
Their $\theta$ difference is less than a fifth of $1^\circ$.
Eigen-based binary feature amalgamation in multimodal biometrics
by Wen-Shiung Chen; Ren-He Jeng
International Journal of Biometrics (IJBM), Vol. 12, No. 3, 2020
Abstract: In this paper, a quantised eigen analysis (QEA) for the extracted features is proposed and an associated eigen-based binary feature amalgamation (EBFA) based on QEA is developed for feature fusion in multimodal biometrics. As opposed to feature combination, EBFA projects heterogeneous features onto the projection kernel and uses only the sign parts to encode the features as bit strings to maximise its expression rather than directly combine them. Thus, the feature codes can be simply concatenated or compared by XOR bit-wise operation into a serial or parallel amalgamated feature vector. To evaluate the performance of EBFA, a series of experiments are performed on multiple biometric modalities, including face, palm-print and iris. The experimental results show that the proposed binary feature amalgamation scheme at feature-level is superior to some other feature fusion methods and score-level methods in terms of multimodal recognition accuracy performance.Online publication date: Tue, 14-Jul-2020