Measure of matrix similarity

You can suggest to use the Wasserstein metric. It is also called the task of the excavator (Earth Movers Distance - EMD). In the imagination, you can imagine the matrix, like piles of earth, which are laid out in a line. The task of the excavator is to equalize the two matrices, that is, to drag the land between the heaps so as to level the ground in both matrices. EMD is used, for example, to compare the histograms of two images in order to quickly say how 2 images are similar - and there are no artificial intelligences to you here. EMD is not the most trivial algorithm and there are various options for its implementation and unfortunately, the Java implementation is one of the most confusing that I have seen. github Here is an example from library tests:

 import com.telmomenezes.jfastemd.Feature2D; import com.telmomenezes.jfastemd.JFastEMD; import com.telmomenezes.jfastemd.Signature; public class Main { static double[] a0 = {1.0, 0.0, 0.0, 0.0}; static double[] a1 = {0.0, 1.0, 0.0, 0.0}; static double[] a2 = {0.0, 1.0, 1.0, 0.0}; static double[] b0 = {1.0, 0.31350830458876927, 0.475451529763324, 0.710099174235318, 0.8180547959863713, 0.8501705482451378, 0.7091117393023645, 0.3421407576224318, 0.0, 0.0, 0.8648715755286225, 0.0, 0.0, 0.2296406823738588, 0.32854154105225764, 0.41240916326716803, 0.2556550109727834, 0.0, 0.0, 0.0, 0.9688367289608703, 0.0, 0.15675415229438464, 0.2556550109727834, 0.4473841617389759, 0.5493536070363795, 0.4274423997306564, 0.0, 0.0, 0.0, 0.9897035895139639, 0.0, 0.0, 0.3659766656118061, 0.49450429339199403, 0.6613006436919866, 0.5263432584090552, 0.0, 0.0, 0.0, 0.9501732176368471, 0.0989008586783988, 0.0989008586783988, 0.3956034347135952, 0.5865550152810103, 0.710099174235318, 0.6355631023347673, 0.2556550109727834, 0.0, 0.0, 0.8570091629463844, 0.0, 0.0, 0.4042535848157063, 0.5841965520250411, 0.7250339741455023, 0.6900589756736945, 0.3421407576224318, 0.0, 0.0, 0.6999031392570128, 0.0, 0.15675415229438464, 0.2776498124065264, 0.5399424749792293, 0.6234493289150748, 0.6160355170421022, 0.31350830458876927, 0.0, 0.0, 0.434403964700911, 0.0, 0.0, 0.0, 0.3863948346682434, 0.434403964700911, 0.2776498124065264, 0.0989008586783988, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0}; static double[] b1 = {0.7690558279672058, 0.5431277903401206, 0.7400048870218827, 0.9431329651179853, 1.0, 0.8608371807886335, 0.6014104886149799, 0.0, 0.0, 0.0, 0.3620851935600804, 0.0, 0.0, 0.0, 0.0, 0.1810425967800402, 0.1810425967800402, 0.0, 0.0, 0.0, 0.8073470804011144, 0.2869457269295445, 0.3620851935600804, 0.3620851935600804, 0.6014104886149799, 0.5082508251725361, 0.0, 0.0, 0.0, 0.0, 0.8407357836698793, 0.1810425967800402, 0.5082508251725361, 0.6892934219525761, 0.7824530853950201, 0.626304483621074, 0.42036789183493967, 0.0, 0.0, 0.0, 0.8189574032199598, 0.42036789183493967, 0.649030920489625, 0.7241703871201608, 0.7241703871201608, 0.8300735172696652, 0.2869457269295445, 0.0, 0.0, 0.0, 0.8703360187326165, 0.42036789183493967, 0.5082508251725361, 0.7073136187644842, 0.921047483801923, 0.7549340506391291, 0.626304483621074, 0.0, 0.0, 0.0, 0.573891453859089, 0.3620851935600804, 0.46798832370958465, 0.5082508251725361, 0.3620851935600804, 0.5431277903401206, 0.1810425967800402, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0}; static double[] b2 = {0.8430654934475055, 0.6239426073894736, 0.7730847346217581, 0.9313448972142149, 1.0, 0.8783820744571478, 0.5344908771065096, 0.1686130986895011, 0.0, 0.0, 0.6419699112188565, 0.0, 0.5058392960685033, 0.604471635932257, 0.5833054849228712, 0.5058392960685033, 0.1686130986895011, 0.0, 0.0, 0.0, 0.3372261973790022, 0.1686130986895011, 0.0, 0.5058392960685033, 0.43585853724275586, 0.43585853724275586, 0.2672454385532548, 0.0, 0.0, 0.0, 0.7287336883921703, 0.43585853724275586, 0.1686130986895011, 0.5833054849228712, 0.5601205897026693, 0.47335681252935546, 0.3372261973790022, 0.0, 0.0, 0.0, 0.7730847346217581, 0.43585853724275586, 0.5058392960685033, 0.5344908771065096, 0.6587529295664228, 0.6239426073894736, 0.1686130986895011, 0.0, 0.0, 0.0, 0.7830149820263362, 0.3915074910131681, 0.43585853724275586, 0.7162562210501102, 0.6587529295664228, 0.6891997754414121, 0.2672454385532548, 0.1686130986895011, 0.0, 0.0, 0.5344908771065096, 0.0, 0.3372261973790022, 0.2672454385532548, 0.3915074910131681, 0.43585853724275586, 0.1686130986895011, 0.0, 0.0, 0.0, 0.1686130986895011, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0}; static double getValue(double[] map, int x, int y, int bins) { return map[(y * bins) + x]; } static Signature getSignature(double[] map, int bins) { // find number of entries in the sparse matrix int n = 0; for (int x = 0; x < bins; x++) { for (int y = 0; y < bins; y++) { if (getValue(map, x, y, bins) > 0) { n++; } } } // compute features and weights Feature2D[] features = new Feature2D[n]; double[] weights = new double[n]; int i = 0; for (int x = 0; x < bins; x++) { for (int y = 0; y < bins; y++) { double val = getValue(map, x, y, bins); if (val > 0) { Feature2D f = new Feature2D(x, y); features[i] = f; weights[i] = val; i++; } } } Signature signature = new Signature(); signature.setNumberOfFeatures(n); signature.setFeatures(features); signature.setWeights(weights); return signature; } static double emdDist(double[] map1, double[] map2, int bins) { Signature sig1 = getSignature(map1, bins); Signature sig2 = getSignature(map2, bins); double dist = JFastEMD.distance(sig1, sig2, 0); return dist; } public static void main(String[] args) { System.out.println("test 1: " + emdDist(a0, a0, 2) + " [expected: 0.0]"); System.out.println("test 1: " + emdDist(a0, a1, 2) + " [expected: 1.0]"); System.out.println("test 2: " + emdDist(a0, a2, 2) + " [expected: 2.0]"); System.out.println("test 3: " + emdDist(b0, b1, 10) + " [expected: 19.1921]"); System.out.println("test 4: " + emdDist(b0, b2, 10) + " [expected: 25.7637]"); } } 

Addition: EMD is proposed not because this method compares any two vectors or matrices (it doesn’t work that way, does not work for all cases!), But because the author explicitly indicated that the matrices consist of some TF-like measure, and EMD is only suitable for vectors containing a probability distribution.

The simplest option is the sum of the squared differences of the corresponding elements. Or the normalized difference. Something like Frobenius norm for matrix difference.