A regression problem is one where the goal is to predict a single numeric value. Common machine learning regression techniques are linear regression, k-nearest neighbors, kernel ridge, Gaussian process, decision tree, AdaBoost (includes bagging), gradient boost, and kernel ridge regression (KRR). Less common regression techniques include random neighborhoods regression, naive Bayes regression, Poisson regression, and a few others.
KRR is a powerful technique that uses linear regression with the kernel trick to deal with complex non-linear data, combined with the ridge technique to discourage model overfitting.
The standard way to train a KRR model is to use an algorithm that involves matrix inversion. In KRR there is one model weight for every training item and if there are n training items, the matrix to invert has size nxn. Therefore, a disadvantage of KRR is that the standard training technique doesn’t scale well to problems with very large datasets. (It’s possible to train a KRR model using stochastic gradient descent but that’s another topic.)
One rainy Pacific Northwest Sunday morning, I decided to refactor my standard C# implementation of KRR. My standard implementation uses the Cholesky decomposition algorithm to compute the matrix inverse of the kernel matrix during training. The Cholesky algorithm is a specialized technique that only works for simple matrices that are symmetric and positive definite — and a kernel matrix meets this requirement. Just for fun, I figured I’d use Newton iteration to compute the inverse of the kernel matrix. Newton iteration is robust and can deal with large (but not huge) matrices, but because Newton is iterative, it’s a bit slow.
For my implementation kernel function, I hard-coded a radial basis function (RBF) where rbf(x1, x2) = exp(-gamma * ||x1 – x2||). There are several different versions of RBF and this can cause a lot of confusion. To recap, when using kernel ridge regression, you must specify which of several kernel functions to use, and the kernel function’s parameters, and when training KRR, you have several techniques to choose from.
The refactored KRR implementation worked quite well. The output of the demo is:
Begin Kernel Ridge Regression with Newton iteration 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 = 0.1 Setting alpha noise = 0.001 Create and train KRR model Done Computing model accuracy Train acc (within 0.10) = 0.9700 Test acc (within 0.10) = 0.9500 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 predicted y = 0.4881 End demo
The demo data is synthetic. There are five predictor variables with values between -1 and +1. The target values to predict are between 0.0 and 1.0. The synthetic data was generated by a 5-10-1 neural network with random weights and biases. Therefore, the data has an underlying, but complex, structure which can, in theory, be predicted. There are 200 training items and 40 test items.
One version of the key equations for kernel ridge regression. In my opinion, the information about KRR on the Internet is among the most ambiguous, confusing, and contradictory for any machine learning topic. One difficulty is that there are dozens of equations that appear quite different but in fact are equivalent. In this diagram, the alpha values are the model weights, the K is the kernel matrix, the phi are the kernel function values, the lambda is the noise regularization, and d(x) is the predicted y value for an input x. This is a screenshot from youtube.com/watch?v=JQJVA8ehlbM.
RBF is a good general purpose kernel function. The values of gamma (0.10) for the RBF function, and noise/alpha (0.001) were determined by trial and error.
The demo program uses the trained KRR model to predict for x = (-0.1660, 0.4406, -0.9998, -0.3953, -0.7065) which is the first training item. The predicted y value is 0.4881 which is quite close to the actual y value of 0.4840.
Kernel ridge regression is closely related to Gaussian process regression. The two techniques start with very different math assumptions but end up with very similar training and prediction equations. The main difference between the techniques is that KRR gives a single point prediction, but GPR gives a point prediction and its standard deviation.
This was an interesting and fun exploration.
In machine learning, a kernel function maps a linear function to a non-linear function — sort like a door from 1-D space to n-D space. Some of my favorite doors in science fiction movies have dilating, or iris, designs.
Left: In “Star Wars: The Empire Strikes Back” (1980), Luke is dangling from Cloud City but is rescued by Lando who pulls him into the Millenium Falcon via a dilating hatch in the roof. My grade for the movie = B.
Center: In “The Atomic Submarine” (1959), in the near future, the crew of the USS Tigershark submarine encounters a submerged alien spacecraft with an unfriendly occupant. A small boarding party manages to get inside the alien craft. Several of the exploring crew die gruesome deaths, including one who is cut in half while trying to escape through a dilating door. The alien craft is destroyed in the end. My grade for the movie = B.
Right: In “Alien Romulus” (2024), a group of young mining workers on planet LV-410 use their mining shuttle to travel to the abandoned space station Renaissance to go to a nice planet Yvaga III. The entrance door to the space station has a dilated design. Only one person and one android manage to survive the aliens on board. My grade for the movie = B.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte”, “and” with Boolean operator symbols. My blog editor chokes on symbols.
using System;
using System.IO;
using System.Collections.Generic;
namespace KernelRidgeRegressionNewton
{
internal class KRRNewtonProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin Kernel Ridge " +
"Regression with Newton iteration matrix inverse ");
// 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 = 0.1; // RBF param
double alpha = 0.001; // regularization
Console.WriteLine("\nSetting RBF gamma = " +
gamma.ToString("F1"));
Console.WriteLine("Setting alpha noise = " +
alpha.ToString("F3"));
Console.WriteLine("\nCreate and train KRR model ");
KRR krr = new KRR(gamma, alpha);
krr.Train(trainX, trainY);
Console.WriteLine("Done ");
// 3. evaluate
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"));
// Console.WriteLine("Model weights: "); // 1 per x
// MatUtils.VecShow(krr.wts, 4, 9);
// 4. use model
double[] x = trainX[0];
Console.WriteLine("\nPredicting for x = ");
MatUtils.VecShow(x, 4, 9);
double predY = krr.Predict(x);
Console.WriteLine("\npredicted y = " +
predY.ToString("F4"));
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main()
} // class 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.MatInverseNewton(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
}
// ------------------------------------------------------
} // 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[][] mat)
{
int nRows = mat.Length;
int nCols = mat[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++] = mat[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] VecToMat(double[] vec,
int nRows, int nCols)
{
double[][] result = MatMake(nRows, nCols);
int k = 0;
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = vec[k++];
return result;
}
// ------------------------------------------------------
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[][] MatProduct(double[][] matA,
double[][] matB)
{
int aRows = matA.Length;
int aCols = matA[0].Length;
int bRows = matB.Length;
int bCols = matB[0].Length;
if (aCols != bRows)
throw new Exception("Non-conformable matrices");
double[][] result = MatMake(aRows, bCols);
for (int i = 0; i "lt" aRows; ++i) // each row of A
for (int j = 0; j "lt" bCols; ++j) // each col of B
for (int k = 0; k "lt" aCols; ++k)
result[i][j] += matA[i][k] * matB[k][j];
return result;
}
// ------------------------------------------------------
public static double[] VecMatProd(double[] v,
double[][] m)
{
// one-dim vec * two-dim mat
int nRows = m.Length;
int nCols = m[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] * m[i][j];
}
}
return result;
}
// ------------------------------------------------------
public static double[][] MatSubtract(double[][] matA,
double[][] matB)
{
// matA - matB
int nRows = matA.Length; int nCols = matA[0].Length;
double[][] result = MatMake(nRows, nCols);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = matA[i][j] - matB[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatTranspose(double[][] m)
{
int nr = m.Length; int nc = m[0].Length;
double[][] result = MatMake(nc, nr); // note
for (int i = 0; i "lt" nr; ++i)
for (int j = 0; j "lt" nc; ++j)
result[j][i] = m[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatCopyOf(double[][] m)
{
int nRows = m.Length; int nCols = m[0].Length;
double[][] result = MatMake(nRows, nCols);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = m[i][j];
return result;
}
// ------------------------------------------------------
public static bool MatAreEqual(double[][] matA,
double[][] matB, double eps)
{
int nr = matA.Length; int nc = matB[0].Length;
for (int i = 0; i "lt" nr; ++i)
for (int j = 0; j "lt" nc; ++j)
if (Math.Abs(matA[i][j] - matB[i][j]) "gt" eps)
return false;
return true;
}
// ------------------------------------------------------
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[][] MatScalarMult(double[][] m,
double u)
{
int nRows = m.Length; int nCols = m[0].Length;
double[][] result = MatMake(nRows, nCols);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = u * m[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatInverseNewton(double[][] A,
int maxIter = 1000, double epsilon = 1.0e-6)
{
// Newton's method
// X_k+1 = X_k * (2I - A*X_k)
int n = A.Length; // must be square martix
double[][] Xprev = NewtonStart(A); // Pan algorithm
double[][] Xnew = MatMake(n, n);
double[][] I = MatIdentity(n);
double[][] I2 = MatScalarMult(I, 2.0);
int iter = 0;
while (iter "lt" maxIter)
{
Xnew =
MatProduct(Xprev, MatSubtract(I2,
MatProduct(A, Xprev)));
Xprev = MatCopyOf(Xnew);
if (iter % 10 == 0)
{
double[][] check = MatProduct(A, Xnew); //
if (MatAreEqual(check, I, epsilon) == true)
return Xnew;
}
++iter;
} // while
Console.WriteLine("WARNING: no matrix inverse converge");
return Xnew;
// ****************************************************
// nested helper: NewtonStart()
// ****************************************************
static double[][] NewtonStart(double[][] m)
{
int n = m.Length;
double maxRowSum = 0.0;
double maxColSum = 0.0;
for (int i = 0; i "lt" n; ++i)
{
double rowSum = 0.0;
for (int j = 0; j "lt" n; ++j)
rowSum += Math.Abs(m[i][j]);
if (rowSum "gt" maxRowSum)
maxRowSum = rowSum;
}
for (int j = 0; j "lt" n; ++j)
{
double colSum = 0.0;
for (int i = 0; i "lt" n; ++i)
colSum += Math.Abs(m[i][j]);
if (colSum "gt" maxColSum)
maxColSum = colSum;
}
double[][] result = MatTranspose(m);
double t = 1.0 / (maxRowSum * maxColSum);
for (int i = 0; i "lt" m.Length; ++i)
for (int j = 0; j "lt" m.Length; ++j)
result[i][j] *= t;
return result;
} // NewtonStart()
// ****************************************************
} // MatInverseNewton()
// ------------------------------------------------------
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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0.2215 -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, 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-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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