Kernel ridge regression (KRR) is a technique to predict a single numeric value. KRR uses a kernel function, which compares two training item vectors and computes a measure of their similarity in order to handle complex non-linear data, and the ridge technique (aka L2 regularization) to prevent model overfitting.
For small and medium sized datasets, kernel ridge regression is often my preferred technique. KRR is complex but when it works, it often produces very accurate predictions.
If you have three xi training vectors, the predicted value for an input vector x is y’ = (w0 * k(x, x0)) + (w1 * k(x, x1)) + (w2 * k(x, x2)). The wi are the model weights, the xi are the predictors, and k() is a kernel function. Of course, you usually have a lot more than just three training items.
There are many possible kernel functions. The most common, based on my experience, is the radial basis function (RBF). There are two main versions of RBF. The one I use is rbf(v1, v2) = exp(-1 * gamma * ||v1 – v2||^2) where gamma is a free parameter, sometimes called the inverse bandwidth, and ||v1 – v2||^2 is the squared Euclidean distance between vectors v1 and v2. If v1 = v2, the RBF value is 1.0 (maximum similarity). As the difference between v1 and v2 increases, RBF decreases towards 0.0.
Training a KRR prediction model is the process of finding the values of the weights. There is one weight per training item. There are several techniques that can be used to train a KRR model. The techniques fall into two categories: 1.) closed-form techniques that use a matrix inverse, and 2.) iterative techniques that do not use a matrix inverse. This blog describes the closed-form training technique that uses a matrix inverse. The technique is fast but does not work for huge datasets.
Training using matrix inverse starts by computing a kernel matrix K which is the kernel measure of similarity between all pairs of training items. If you have n training items, the K matrix is n-by-n. K[0][0] holds the RBF measure of similarity between training item [0] and itself, which will be 1.0 because you’re comparing a vector with itself. K[3][5] holds the RBF measure of similarity between training items [3] and [5]. K[5][3] holds the RBF measure of similarity between training items [5] and [3], which will be the same as K[3][5]. So, the K matrix will be a n-by-n, symmetric matrix, with positive values, initially with 1.0 values on the diagonal.
The “ridge” part of KRR adds a small value alpha, usually about 0.001 or so, to the 1.0 values on the diagonal of the K matrix. This helps to prevent model overfitting, when the model weights are too good and predict the training data perfectly, but then the model predicts poorly on new, previously unseen data. Adding alpha to the diagonal of the K matrix also “conditions” it so that computing the matrix inverse of K is less likely to fail (all matrix inverse techniques are extremely brittle and can fail for many reasons).
The model weights are computed by w = Y * inv(K(X, X)) where w is a vector of the weights, Y is a 1-by-n column-matrix of all target training values, and inv(K(X, X)) is the inverse of the kernel matrix. There are several ways to compute a matrix inverse. General matrix inverse techniques include LUP inverse, SVD inverse, and QR inverse. But because the K kernel matrix is symmetric with positive values, there’s a specialized inverse technique, called Cholesky decomposition inverse, that can be used.
For my demo, I used synthetic data that looks like:
-0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 . . .
The first five values on each line are predictor values. The last value is the target y value to predict. The data was generated by a 5-10-1 neural network with random weights and bias values. There are 200 training items and 40 test items.
The key calling statements in my demo are:
. . .
double gamma = 0.3; // RBF param
double alpha = 0.005; // regularization
KRR krr = new KRR(gamma, alpha);
krr.Train(trainX, trainY);
double trainAcc = krr.Accuracy(trainX, trainY, 0.10);
double testAcc = krr.Accuracy(testX, testY, 0.10);
double trainMSE = krr.MSE(trainX, trainY);
double testMSE = krr.MSE(testX, testY);
. . .
double[] x = trainX[0]; // predict the first train item
double predY = krr.Predict(x);
Console.WriteLine("Predicted y = " +
predY.ToString("F4"));
The Accuracy() method scores a prediction as correct if it’s within 10% of the true target y value. MSE is mean squared error, which is a more granular measure of model effectivenesss than accuracy.
The output of my demo is:
Begin Kernel Ridge Regression with Cholesky matrix inverse Loading train (200) and test (40) from file Done First three train X: -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, First three train y: 0.4840 0.1568 0.8054 Setting RBF gamma = 0.3 Setting alpha noise = 0.005 Creating and training KRR model using Cholesky Done Model weights: -2.0218 -1.1406 0.0758 -0.6265 . . . 0.3933 0.2223 0.0564 0.4282 . . . . . . -0.2014 -1.6270 -0.5825 -0.0487 . . . Computing model accuracy Train acc (within 0.10) = 0.9950 Test acc (within 0.10) = 0.9500 Train MSE = 0.0000 Test MSE = 0.0002 Predicting for x = -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, Predicted y = 0.4941 End demo
From a practical point of view, the main challenge is to find good values for the gamma parameter (that controls the RBF kernel similarity function) and the alpha noise parameter (that prevents model overfitting).
I really enjoy implementing machine learning algorithms. The process is a bit like solving a mystery.
I love mystery movies, but sadly, there are very few super good ones. By good ones, I mean mystery movies where there are muliple suspects, each suspect has a plausible motive, and there are clues that help you as the viewer to correctly guess the culprit. If you’re watching a movie like this with a group of friends, and people are saying things like, “My money is on the banker. His divorce made him poor and he wants revenge.” — then that’s the kind of movie I like.
Left: I recently watched one of the best mystery moivies I’ve seen in a long time. In “Drop” (2025), a woman does on a blind dinner date. She starts getting text messages from a phone application, where the sender must be someone in the restaurant. The messages tell her that she must murder her date using poison, or her young son at home will die.
The suspects include a random guy she literally bumped into twice (accidentally?), an overly-friendly bartender, the newly-hired piano player, an older man who is also nervously waiting for a blind date, the restaurant hostess who seated the couple at a very specific table, an annoying waiter, and a group of prom kids. I didn’t like the lead actress — her character came across a bit like an entitled witch in a Lifetime TV movie — but the story and all the other actor portrayals were excellent.
Right: One of the most famous, and arguably best, mysteries of all time is “Murder on the Orient Express” (1974). It’s based on the 1934 novel of the same name by Agatha Christie. A ruthless business man is murdered on a snowbound train. Belgian detective Hercule Poirot must figure out which of the 15 passengers is the murderer. A-list actors, A-grade production values, and an A-grade plot.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols (my blog editor usually chokes on symbols).
using System;
using System.IO;
using System.Collections.Generic;
namespace KernelRidgeRegressionCholesky
{
internal class KRRCholeskyProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin Kernel Ridge " +
"Regression with Cholesky matrix inverse training ");
// 1. Load data
Console.WriteLine("\nLoading train (200) and" +
" test (40) from file ");
string trainFile =
"..\\..\\..\\Data\\synthetic_train_200.txt";
double[][] trainX = MatUtils.MatLoad(trainFile,
new int[] { 0, 1, 2, 3, 4 }, ',', "#");
double[] trainY =
MatUtils.MatToVec(MatUtils.MatLoad(trainFile,
new int[] { 5 }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX = MatUtils.MatLoad(testFile,
new int[] { 0, 1, 2, 3, 4 }, ',', "#");
double[] testY =
MatUtils.MatToVec(MatUtils.MatLoad(testFile,
new int[] { 5 }, ',', "#"));
Console.WriteLine("Done ");
Console.WriteLine("\nFirst three train X: ");
for (int i = 0; i "lt" 3; ++i)
MatUtils.VecShow(trainX[i], 4, 8);
Console.WriteLine("\nFirst three train y: ");
for (int i = 0; i "lt" 3; ++i)
Console.WriteLine(trainY[i].ToString("F4").
PadLeft(8));
// 2. create and train KRR model
double gamma = 0.3; // RBF param
double alpha = 0.005; // regularization
Console.WriteLine("\nSetting RBF gamma = " +
gamma.ToString("F1"));
Console.WriteLine("Setting alpha noise = " +
alpha.ToString("F3"));
Console.WriteLine("\nCreating and training KRR" +
" model using Cholesky ");
KRR krr = new KRR(gamma, alpha);
krr.Train(trainX, trainY);
Console.WriteLine("Done ");
Console.WriteLine("\nModel weights: ");
MatUtils.VecShow(krr.wts, 4, 9);
// Console.WriteLine("\nAnalysis of model weights: ");
// MatUtils.VecAnalyze(krr.wts);
// 3. evaluate model
Console.WriteLine("\nComputing model accuracy ");
double trainAcc = krr.Accuracy(trainX, trainY, 0.10);
double testAcc = krr.Accuracy(testX, testY, 0.10);
Console.WriteLine("\nTrain acc (within 0.10) = " +
trainAcc.ToString("F4"));
Console.WriteLine("Test acc (within 0.10) = " +
testAcc.ToString("F4"));
double trainMSE = krr.MSE(trainX, trainY);
double testMSE = krr.MSE(testX, testY);
Console.WriteLine("\nTrain MSE = " +
trainMSE.ToString("F4"));
Console.WriteLine("Test MSE = " +
testMSE.ToString("F4"));
// 4. use model
double[] x = trainX[0];
Console.WriteLine("\nPredicting for x = ");
MatUtils.VecShow(x, 4, 9);
double predY = krr.Predict(x);
Console.WriteLine("Predicted y = " +
predY.ToString("F4"));
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main()
} // Program
// ========================================================
public class KRR
{
public double gamma; // for RBF kernel
public double alpha; // regularization noise
public double[][] trainX; // need for any prediction
public double[] trainY; // not necessary
public double[] wts; // one per trainX item
// ------------------------------------------------------
public KRR(double gamma, double alpha) // ctor()
{
this.gamma = gamma;
this.alpha = alpha;
}
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
// 0. store trainX -- needed by Predict()
this.trainX = trainX; // by ref -- could copy
this.trainY = trainY; // not used this version
// 1. compute train-train K (covariance) matrix
int N = trainX.Length;
double[][] K = MatUtils.MatMake(N, N);
for (int i = 0; i "lt" N; ++i)
for (int j = 0; j "lt" N; ++j)
K[i][j] = Rbf(trainX[i], trainX[j], this.gamma);
// 2. add regularization on diagonal
for (int i = 0; i "lt" N; ++i)
K[i][i] += this.alpha;
// 3. compute model weights using K inverse
double[][] Kinv =
MatUtils.MatInverseCholesky(K);
this.wts = MatUtils.VecMatProd(trainY, Kinv);
} // Train
// ------------------------------------------------------
public double Predict(double[] x)
{
int N = this.trainX.Length;
double sum = 0.0;
for (int i = 0; i "lt" N; ++i)
{
double[] xx = this.trainX[i];
double k = Rbf(x, xx, this.gamma);
sum += this.wts[i] * k;
}
return sum;
}
// ------------------------------------------------------
public double Accuracy(double[][] dataX, double[] dataY,
double pctClose)
{
int numCorrect = 0; int numWrong = 0;
int n = dataX.Length;
for (int i = 0; i "lt" n; ++i)
{
double[] x = dataX[i];
double predY = this.Predict(x);
double actualY = dataY[i];
if (Math.Abs(predY - actualY)
"lt" Math.Abs(pctClose * actualY))
numCorrect += 1;
else
numWrong += 1;
}
return (numCorrect * 1.0) / (numCorrect + numWrong);
}
// ------------------------------------------------------
private static double Rbf(double[] v1, double[] v2,
double gamma)
{
// the gamma version aot len_scale version
int dim = v1.Length;
double sum = 0.0;
for (int i = 0; i "lt" dim; ++i)
{
sum += (v1[i] - v2[i]) * (v1[i] - v2[i]);
}
return Math.Exp(-1 * gamma * sum); // before
}
// ------------------------------------------------------
public double MSE(double[][] dataX, double[] dataY)
{
double sum = 0.0;
int n = dataX.Length;
for (int i = 0; i "lt" n; ++i)
{
double[] x = dataX[i];
double actualY = dataY[i];
double predY = this.Predict(x);
sum += (actualY - predY) * (actualY - predY);
}
return sum / n;
}
} // class KRR
// ========================================================
public class MatUtils
{
// ------------------------------------------------------
public static double[][] MatLoad(string fn,
int[] usecols, char sep, string comment)
{
List"lt"double[]"gt" result = new List"lt"double[]"gt"();
string line = "";
FileStream ifs = new FileStream(fn, FileMode.Open);
StreamReader sr = new StreamReader(ifs);
while ((line = sr.ReadLine()) != null)
{
if (line.StartsWith(comment) == true)
continue;
string[] tokens = line.Split(sep);
List"lt"double"gt" lst = new List"lt"double"gt"();
for (int j = 0; j "lt" usecols.Length; ++j)
lst.Add(double.Parse(tokens[usecols[j]]));
double[] row = lst.ToArray();
result.Add(row);
}
sr.Close(); ifs.Close();
return result.ToArray();
}
// ------------------------------------------------------
public static double[] MatToVec(double[][] A)
{
int nRows = A.Length;
int nCols = A[0].Length;
double[] result = new double[nRows * nCols];
int k = 0;
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[k++] = A[i][j];
return result;
}
// ------------------------------------------------------
public static void VecAnalyze(double[] vec)
{
// to programmatically analyze trained model weights
int n = vec.Length;
double smallest = vec[0];
double largest = vec[0];
double sum = 0.0;
for (int i = 0; i "lt" n; ++i)
{
if (vec[i] "lt" smallest)
smallest = vec[i];
if (vec[i] "gt" largest)
largest = vec[i];
sum += Math.Abs(vec[i]);
}
double avg = sum / n;
Console.WriteLine("Smallest value = " +
smallest.ToString("F4"));
Console.WriteLine("Largest value = " +
largest.ToString("F4"));
Console.WriteLine("Average magnitude = " +
avg.ToString("F4"));
}
// ------------------------------------------------------
public static double[][] MatMake(int nRows, int nCols)
{
double[][] result = new double[nRows][];
for (int i = 0; i "lt" nRows; ++i)
result[i] = new double[nCols];
return result;
}
// ------------------------------------------------------
public static double[] VecMatProd(double[] v,
double[][] A)
{
// one-dim vec * two-dim mat
int nRows = A.Length;
int nCols = A[0].Length;
int n = v.Length;
if (n != nCols)
throw new Exception("non-comform in VecMatProd");
double[] result = new double[n];
for (int i = 0; i "lt" n; ++i)
{
for (int j = 0; j "lt" nCols; ++j)
{
result[i] += v[j] * A[i][j];
}
}
return result;
}
// ------------------------------------------------------
public static double[][] MatIdentity(int n)
{
double[][] result = MatMake(n, n);
for (int i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
return result;
}
// ------------------------------------------------------
public static double[][] MatInverseCholesky(double[][] A)
{
double[][] L = MatDecompCholesky(A);
double[][] result = MatInverseFromCholesky(L);
return result;
// ----------------------------------------------------
// nested helpers:
// MatDecompCholesky, MatInverseFromCholesky
// ----------------------------------------------------
static double[][] MatDecompCholesky(double[][] M)
{
// Cholesky decomposition (Banachiewicz algorithm)
// M is square, symmetric, positive definite
int n = M.Length;
double[][] result = MatMake(n, n); // all 0.0
for (int i = 0; i "lt" n; ++i)
{
for (int j = 0; j "lte" i; ++j)
{
double sum = 0.0;
for (int k = 0; k "lt" j; ++k)
sum += result[i][k] * result[j][k];
if (i == j)
{
double tmp = M[i][i] - sum;
if (tmp "lt" 0.0)
throw new
Exception("MatDecompCholesky fatal");
result[i][j] = Math.Sqrt(tmp);
}
else
{
if (result[j][j] == 0.0)
throw new
Exception("MatDecompCholesky fatal ");
result[i][j] =
(1.0 / result[j][j] * (M[i][j] - sum));
}
} // j
} // i
return result;
} // MatDecompCholesky
static double[][] MatInverseFromCholesky(double[][] L)
{
// L is a lower triangular result of Cholesky decomp
// direct version
int n = L.Length;
double[][] result = MatIdentity(n);
for (int k = 0; k "lt" n; ++k)
{
for (int j = 0; j "lt" n; j++)
{
for (int i = 0; i "lt" k; i++)
{
result[k][j] -= result[i][j] * L[k][i];
}
result[k][j] /= L[k][k];
}
}
for (int k = n - 1; k "gte" 0; --k)
{
for (int j = 0; j "lt" n; j++)
{
for (int i = k + 1; i "lt" n; i++)
{
result[k][j] -= result[i][j] * L[i][k];
}
result[k][j] /= L[k][k];
}
}
return result;
}
}
// ------------------------------------------------------
public static void MatShow(double[][] m, int dec, int wid)
{
int nRows = m.Length; int nCols = m[0].Length;
double small = 1.0 / Math.Pow(10, dec);
for (int i = 0; i "lt" nRows; ++i)
{
for (int j = 0; j "lt" nCols; ++j)
{
double v = m[i][j];
if (Math.Abs(v) "lt" small) v = 0.0;
Console.Write(v.ToString("F" + dec).
PadLeft(wid));
}
Console.WriteLine("");
}
}
// ------------------------------------------------------
public static void VecShow(double[] vec, int dec, int wid)
{
for (int i = 0; i "lt" vec.Length; ++i)
Console.Write(vec[i].ToString("F" + dec).PadLeft(wid));
Console.WriteLine("");
}
// ------------------------------------------------------
// ------------------------------------------------------
} // class MatUtils
// ========================================================
} // ns
Training data:
# synthetic_train_200.txt # -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 0.9365, -0.3732, 0.3846, 0.7528, 0.7892, 0.1345 -0.8299, -0.9219, -0.6603, 0.7563, -0.8033, 0.7955 0.0663, 0.3838, -0.3690, 0.3730, 0.6693, 0.3206 -0.9634, 0.5003, 0.9777, 0.4963, -0.4391, 0.7377 -0.1042, 0.8172, -0.4128, -0.4244, -0.7399, 0.4801 -0.9613, 0.3577, -0.5767, -0.4689, -0.0169, 0.6861 -0.7065, 0.1786, 0.3995, -0.7953, -0.1719, 0.5569 0.3888, -0.1716, -0.9001, 0.0718, 0.3276, 0.2500 0.1731, 0.8068, -0.7251, -0.7214, 0.6148, 0.3297 -0.2046, -0.6693, 0.8550, -0.3045, 0.5016, 0.2129 0.2473, 0.5019, -0.3022, -0.4601, 0.7918, 0.2613 -0.1438, 0.9297, 0.3269, 0.2434, -0.7705, 0.5171 0.1568, -0.1837, -0.5259, 0.8068, 0.1474, 0.3307 -0.9943, 0.2343, -0.3467, 0.0541, 0.7719, 0.5581 0.2467, -0.9684, 0.8589, 0.3818, 0.9946, 0.1092 -0.6553, -0.7257, 0.8652, 0.3936, -0.8680, 0.7018 0.8460, 0.4230, -0.7515, -0.9602, -0.9476, 0.1996 -0.9434, -0.5076, 0.7201, 0.0777, 0.1056, 0.5664 0.9392, 0.1221, -0.9627, 0.6013, -0.5341, 0.1533 0.6142, -0.2243, 0.7271, 0.4942, 0.1125, 0.1661 0.4260, 0.1194, -0.9749, -0.8561, 0.9346, 0.2230 0.1362, -0.5934, -0.4953, 0.4877, -0.6091, 0.3810 0.6937, -0.5203, -0.0125, 0.2399, 0.6580, 0.1460 -0.6864, -0.9628, -0.8600, -0.0273, 0.2127, 0.5387 0.9772, 0.1595, -0.2397, 0.1019, 0.4907, 0.1611 0.3385, -0.4702, -0.8673, -0.2598, 0.2594, 0.2270 -0.8669, -0.4794, 0.6095, -0.6131, 0.2789, 0.4700 0.0493, 0.8496, -0.4734, -0.8681, 0.4701, 0.3516 0.8639, -0.9721, -0.5313, 0.2336, 0.8980, 0.1412 0.9004, 0.1133, 0.8312, 0.2831, -0.2200, 0.1782 0.0991, 0.8524, 0.8375, -0.2102, 0.9265, 0.2150 -0.6521, -0.7473, -0.7298, 0.0113, -0.9570, 0.7422 0.6190, -0.3105, 0.8802, 0.1640, 0.7577, 0.1056 0.6895, 0.8108, -0.0802, 0.0927, 0.5972, 0.2214 0.1982, -0.9689, 0.1870, -0.1326, 0.6147, 0.1310 -0.3695, 0.7858, 0.1557, -0.6320, 0.5759, 0.3773 -0.1596, 0.3581, 0.8372, -0.9992, 0.9535, 0.2071 -0.2468, 0.9476, 0.2094, 0.6577, 0.1494, 0.4132 0.1737, 0.5000, 0.7166, 0.5102, 0.3961, 0.2611 0.7290, -0.3546, 0.3416, -0.0983, -0.2358, 0.1332 -0.3652, 0.2438, -0.1395, 0.9476, 0.3556, 0.4170 -0.6029, -0.1466, -0.3133, 0.5953, 0.7600, 0.4334 -0.4596, -0.4953, 0.7098, 0.0554, 0.6043, 0.2775 0.1450, 0.4663, 0.0380, 0.5418, 0.1377, 0.2931 -0.8636, -0.2442, -0.8407, 0.9656, -0.6368, 0.7429 0.6237, 0.7499, 0.3768, 0.1390, -0.6781, 0.2185 -0.5499, 0.1850, -0.3755, 0.8326, 0.8193, 0.4399 -0.4858, -0.7782, -0.6141, -0.0008, 0.4572, 0.4197 0.7033, -0.1683, 0.2334, -0.5327, -0.7961, 0.1776 0.0317, -0.0457, -0.6947, 0.2436, 0.0880, 0.3345 0.5031, -0.5559, 0.0387, 0.5706, -0.9553, 0.3107 -0.3513, 0.7458, 0.6894, 0.0769, 0.7332, 0.3170 0.2205, 0.5992, -0.9309, 0.5405, 0.4635, 0.3532 -0.4806, -0.4859, 0.2646, -0.3094, 0.5932, 0.3202 0.9809, -0.3995, -0.7140, 0.8026, 0.0831, 0.1600 0.9495, 0.2732, 0.9878, 0.0921, 0.0529, 0.1289 -0.9476, -0.6792, 0.4913, -0.9392, -0.2669, 0.5966 0.7247, 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-0.1628, 0.2002 0.0115, -0.6209, 0.9300, -0.4116, -0.7931, 0.4052 -0.7114, -0.9718, 0.4319, 0.1290, 0.5892, 0.3661 0.3915, 0.5557, -0.1870, 0.2955, -0.6404, 0.2954 -0.3564, -0.6548, -0.1827, -0.5172, -0.1862, 0.4622 0.2392, -0.4959, 0.5857, -0.1341, -0.2850, 0.2470 -0.3394, 0.3947, -0.4627, 0.6166, -0.4094, 0.5325 0.7107, 0.7768, -0.6312, 0.1707, 0.7964, 0.2757 -0.1078, 0.8437, -0.4420, 0.2177, 0.3649, 0.4028 -0.3139, 0.5595, -0.6505, -0.3161, -0.7108, 0.5546 0.4335, 0.3986, 0.3770, -0.4932, 0.3847, 0.1810 -0.2562, -0.2894, -0.8847, 0.2633, 0.4146, 0.4036 0.2272, 0.2966, -0.6601, -0.7011, 0.0284, 0.2778 -0.0743, -0.1421, -0.0054, -0.6770, -0.3151, 0.3597 -0.4762, 0.6891, 0.6007, -0.1467, 0.2140, 0.4266 -0.4061, 0.7193, 0.3432, 0.2669, -0.7505, 0.6147 -0.0588, 0.9731, 0.8966, 0.2902, -0.6966, 0.4955 -0.0627, -0.1439, 0.1985, 0.6999, 0.5022, 0.3077 0.1587, 0.8494, -0.8705, 0.9827, -0.8940, 0.4263 -0.7850, 0.2473, -0.9040, -0.4308, -0.8779, 0.7199 0.4070, 0.3369, -0.2428, -0.6236, 0.4940, 0.2215 -0.0242, 0.0513, -0.9430, 0.2885, -0.2987, 0.3947 -0.5416, -0.1322, -0.2351, -0.0604, 0.9590, 0.3683 0.1055, 0.7783, -0.2901, -0.5090, 0.8220, 0.2984 -0.9129, 0.9015, 0.1128, -0.2473, 0.9901, 0.4776 -0.9378, 0.1424, -0.6391, 0.2619, 0.9618, 0.5368 0.7498, -0.0963, 0.4169, 0.5549, -0.0103, 0.1614 -0.2612, -0.7156, 0.4538, -0.0460, -0.1022, 0.3717 0.7720, 0.0552, -0.1818, -0.4622, -0.8560, 0.1685 -0.4177, 0.0070, 0.9319, -0.7812, 0.3461, 0.3052 -0.0001, 0.5542, -0.7128, -0.8336, -0.2016, 0.3803 0.5356, -0.4194, -0.5662, -0.9666, -0.2027, 0.1776 -0.2378, 0.3187, -0.8582, -0.6948, -0.9668, 0.5474 -0.1947, -0.3579, 0.1158, 0.9869, 0.6690, 0.2992 0.3992, 0.8365, -0.9205, -0.8593, -0.0520, 0.3154 -0.0209, 0.0793, 0.7905, -0.1067, 0.7541, 0.1864 -0.4928, -0.4524, -0.3433, 0.0951, -0.5597, 0.6261 -0.8118, 0.7404, -0.5263, -0.2280, 0.1431, 0.6349 0.0516, -0.8480, 0.7483, 0.9023, 0.6250, 0.1959 -0.3212, 0.1093, 0.9488, -0.3766, 0.3376, 0.2735 -0.3481, 0.5490, -0.3484, 0.7797, 0.5034, 0.4379 -0.5785, -0.9170, -0.3563, -0.9258, 0.3877, 0.4121 0.3407, -0.1391, 0.5356, 0.0720, -0.9203, 0.3458 -0.3287, -0.8954, 0.2102, 0.0241, 0.2349, 0.3247 -0.1353, 0.6954, -0.0919, -0.9692, 0.7461, 0.3338 0.9036, -0.8982, -0.5299, -0.8733, -0.1567, 0.1187 0.7277, -0.8368, -0.0538, -0.7489, 0.5458, 0.0830 0.9049, 0.8878, 0.2279, 0.9470, -0.3103, 0.2194 0.7957, -0.1308, -0.5284, 0.8817, 0.3684, 0.2172 0.4647, -0.4931, 0.2010, 0.6292, -0.8918, 0.3371 -0.7390, 0.6849, 0.2367, 0.0626, -0.5034, 0.7039 -0.1567, -0.8711, 0.7940, -0.5932, 0.6525, 0.1710 0.7635, -0.0265, 0.1969, 0.0545, 0.2496, 0.1445 0.7675, 0.1354, -0.7698, -0.5460, 0.1920, 0.1728 -0.5211, -0.7372, -0.6763, 0.6897, 0.2044, 0.5217 0.1913, 0.1980, 0.2314, -0.8816, 0.5006, 0.1998 0.8964, 0.0694, -0.6149, 0.5059, -0.9854, 0.1825 0.1767, 0.7104, 0.2093, 0.6452, 0.7590, 0.2832 -0.3580, -0.7541, 0.4426, -0.1193, -0.7465, 0.5657 -0.5996, 0.5766, -0.9758, -0.3933, -0.9572, 0.6800 0.9950, 0.1641, -0.4132, 0.8579, 0.0142, 0.2003 -0.4717, -0.3894, -0.2567, -0.5111, 0.1691, 0.4266 0.3917, -0.8561, 0.9422, 0.5061, 0.6123, 0.1212 -0.0366, -0.1087, 0.3449, -0.1025, 0.4086, 0.2475 0.3633, 0.3943, 0.2372, -0.6980, 0.5216, 0.1925 -0.5325, -0.6466, -0.2178, -0.3589, 0.6310, 0.3568 0.2271, 0.5200, -0.1447, -0.8011, -0.7699, 0.3128 0.6415, 0.1993, 0.3777, -0.0178, -0.8237, 0.2181 -0.5298, -0.0768, -0.6028, -0.9490, 0.4588, 0.4356 0.6870, -0.1431, 0.7294, 0.3141, 0.1621, 0.1632 -0.5985, 0.0591, 0.7889, -0.3900, 0.7419, 0.2945 0.3661, 0.7984, -0.8486, 0.7572, -0.6183, 0.3449 0.6995, 0.3342, -0.3113, -0.6972, 0.2707, 0.1712 0.2565, 0.9126, 0.1798, -0.6043, -0.1413, 0.2893 -0.3265, 0.9839, -0.2395, 0.9854, 0.0376, 0.4770 0.2690, -0.1722, 0.9818, 0.8599, -0.7015, 0.3954 -0.2102, -0.0768, 0.1219, 0.5607, -0.0256, 0.3949 0.8216, -0.9555, 0.6422, -0.6231, 0.3715, 0.0801 -0.2896, 0.9484, -0.7545, -0.6249, 0.7789, 0.4370 -0.9985, -0.5448, -0.7092, -0.5931, 0.7926, 0.5402
Test data:
# synthetic_test_40.txt # 0.7462, 0.4006, -0.0590, 0.6543, -0.0083, 0.1935 0.8495, -0.2260, -0.0142, -0.4911, 0.7699, 0.1078 -0.2335, -0.4049, 0.4352, -0.6183, -0.7636, 0.5088 0.1810, -0.5142, 0.2465, 0.2767, -0.3449, 0.3136 -0.8650, 0.7611, -0.0801, 0.5277, -0.4922, 0.7140 -0.2358, -0.7466, -0.5115, -0.8413, -0.3943, 0.4533 0.4834, 0.2300, 0.3448, -0.9832, 0.3568, 0.1360 -0.6502, -0.6300, 0.6885, 0.9652, 0.8275, 0.3046 -0.3053, 0.5604, 0.0929, 0.6329, -0.0325, 0.4756 -0.7995, 0.0740, -0.2680, 0.2086, 0.9176, 0.4565 -0.2144, -0.2141, 0.5813, 0.2902, -0.2122, 0.4119 -0.7278, -0.0987, -0.3312, -0.5641, 0.8515, 0.4438 0.3793, 0.1976, 0.4933, 0.0839, 0.4011, 0.1905 -0.8568, 0.9573, -0.5272, 0.3212, -0.8207, 0.7415 -0.5785, 0.0056, -0.7901, -0.2223, 0.0760, 0.5551 0.0735, -0.2188, 0.3925, 0.3570, 0.3746, 0.2191 0.1230, -0.2838, 0.2262, 0.8715, 0.1938, 0.2878 0.4792, -0.9248, 0.5295, 0.0366, -0.9894, 0.3149 -0.4456, 0.0697, 0.5359, -0.8938, 0.0981, 0.3879 0.8629, -0.8505, -0.4464, 0.8385, 0.5300, 0.1769 0.1995, 0.6659, 0.7921, 0.9454, 0.9970, 0.2330 -0.0249, -0.3066, -0.2927, -0.4923, 0.8220, 0.2437 0.4513, -0.9481, -0.0770, -0.4374, -0.9421, 0.2879 -0.3405, 0.5931, -0.3507, -0.3842, 0.8562, 0.3987 0.9538, 0.0471, 0.9039, 0.7760, 0.0361, 0.1706 -0.0887, 0.2104, 0.9808, 0.5478, -0.3314, 0.4128 -0.8220, -0.6302, 0.0537, -0.1658, 0.6013, 0.4306 -0.4123, -0.2880, 0.9074, -0.0461, -0.4435, 0.5144 0.0060, 0.2867, -0.7775, 0.5161, 0.7039, 0.3599 -0.7968, -0.5484, 0.9426, -0.4308, 0.8148, 0.2979 0.7811, 0.8450, -0.6877, 0.7594, 0.2640, 0.2362 -0.6802, -0.1113, -0.8325, -0.6694, -0.6056, 0.6544 0.3821, 0.1476, 0.7466, -0.5107, 0.2592, 0.1648 0.7265, 0.9683, -0.9803, -0.4943, -0.5523, 0.2454 -0.9049, -0.9797, -0.0196, -0.9090, -0.4433, 0.6447 -0.4607, 0.1811, -0.2389, 0.4050, -0.0078, 0.5229 0.2664, -0.2932, -0.4259, -0.7336, 0.8742, 0.1834 -0.4507, 0.1029, -0.6294, -0.1158, -0.6294, 0.6081 0.8948, -0.0124, 0.9278, 0.2899, -0.0314, 0.1534 -0.1323, -0.8813, -0.0146, -0.0697, 0.6135, 0.2386
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