17 files changed, 22808 insertions, 199 deletions
diff --git a/Makefile b/Makefile
index 45c7329..1eccaac 100644
--- a/Makefile
+++ b/Makefile
@@ -1,7 +1,10 @@
-all: poly abs
+all: fits fits_abs many_over
 
-poly:
+fits: fits.cpp fits.hpp
 	clang++ -std=c++17 -Wno-mathematical-notation-identifier-extension -O3 -march=native -mtune=native -fopenmp fits.cpp -o fits
 
-abs:
+many_over: many_over.cpp fits.hpp
+	clang++ -std=c++17 -Wno-mathematical-notation-identifier-extension -O3 -march=native -mtune=native -fopenmp many_over.cpp -o many_over
+
+fits_abs: fits.cpp fits.hpp
 	clang++ -std=c++17 -Wno-mathematical-notation-identifier-extension -O3 -march=native -mtune=native -fopenmp -DFITS_ABS_BASIS fits.cpp -o fits_abs
diff --git a/figs/fit_overparamfit_poly_1.pdf b/figs/fit_overparamfit_poly_1.pdf
new file mode 100644
index 0000000..8acf5d8
--- /dev/null
+++ b/figs/fit_overparamfit_poly_1.pdf
diff --git a/figs/fit_overparamfit_poly_10.pdf b/figs/fit_overparamfit_poly_10.pdf
new file mode 100644
index 0000000..9272c8d
--- /dev/null
+++ b/figs/fit_overparamfit_poly_10.pdf
diff --git a/figs/fit_overparamfit_poly_2.pdf b/figs/fit_overparamfit_poly_2.pdf
new file mode 100644
index 0000000..3e458c3
--- /dev/null
+++ b/figs/fit_overparamfit_poly_2.pdf
diff --git a/figs/fit_overparamfit_poly_3.pdf b/figs/fit_overparamfit_poly_3.pdf
new file mode 100644
index 0000000..d947b83
--- /dev/null
+++ b/figs/fit_overparamfit_poly_3.pdf
diff --git a/figs/fit_overparamfit_poly_4.pdf b/figs/fit_overparamfit_poly_4.pdf
new file mode 100644
index 0000000..acd4d4f
--- /dev/null
+++ b/figs/fit_overparamfit_poly_4.pdf
diff --git a/figs/fit_overparamfit_poly_5.pdf b/figs/fit_overparamfit_poly_5.pdf
new file mode 100644
index 0000000..342a28b
--- /dev/null
+++ b/figs/fit_overparamfit_poly_5.pdf
diff --git a/figs/fit_overparamfit_poly_6.pdf b/figs/fit_overparamfit_poly_6.pdf
new file mode 100644
index 0000000..41a6d3a
--- /dev/null
+++ b/figs/fit_overparamfit_poly_6.pdf
diff --git a/figs/fit_overparamfit_poly_7.pdf b/figs/fit_overparamfit_poly_7.pdf
new file mode 100644
index 0000000..1f57852
--- /dev/null
+++ b/figs/fit_overparamfit_poly_7.pdf
diff --git a/figs/fit_overparamfit_poly_8.pdf b/figs/fit_overparamfit_poly_8.pdf
new file mode 100644
index 0000000..b8b6f51
--- /dev/null
+++ b/figs/fit_overparamfit_poly_8.pdf
diff --git a/figs/fit_overparamfit_poly_9.pdf b/figs/fit_overparamfit_poly_9.pdf
new file mode 100644
index 0000000..5a5a4c3
--- /dev/null
+++ b/figs/fit_overparamfit_poly_9.pdf
diff --git a/fits.cpp b/fits.cpp
index 09f9be5..682886a 100644
--- a/fits.cpp
+++ b/fits.cpp
@@ -1,145 +1,7 @@
 #include <getopt.h>
 #include <iomanip>
 
-#include "pcg-cpp/include/pcg_random.hpp"
-#include "randutils/randutils.hpp"
-
-#include "eigen/Eigen/Dense"
-
-using Rng = randutils::random_generator<pcg32>;
-
-using Real = double;
-using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
-using Matrix = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
-using Data = std::vector<std::tuple<Real, Real>>;
-
-#ifndef FITS_ABS_BASIS
-inline Real basisFunctions(unsigned N, unsigned i, Real x) {
-  return std::legendre(i, 2.0 * (x - 0.5));
-}
-#endif
-
-#ifdef FITS_ABS_BASIS
-inline Real basisFunctions(unsigned N, unsigned i, Real x) {
-  return abs(x - i / (N - 1.0));
-}
-#endif
-
-Real value(const Vector& coeffs, Real x) {
-  Real v = 0;
-
-  for (unsigned j = 0; j < coeffs.size(); j++) {
-    v += coeffs[j] * basisFunctions(coeffs.size(), j, x);
-  }
-
-  return v;
-}
-
-Real cost(Data::const_iterator batchBegin, Data::const_iterator batchEnd, const Vector& coeffs) {
-  Real c = 0;
-
-#pragma omp parallel for reduction(+:c)
-  for (Data::const_iterator it = batchBegin; it != batchEnd; it++) {
-    Real x = std::get<0>(*it);
-    Real y = std::get<1>(*it);
-    c += 0.5 * pow(value(coeffs, x) - y, 2);
-  }
-
-  return c;
-}
-
-Vector dCost(Data::const_iterator batchBegin, Data::const_iterator batchEnd, const Vector& coeffs) {
-  Vector dC = Vector::Zero(coeffs.size());
-
-  for (Data::const_iterator it = batchBegin; it != batchEnd; it++) {
-    Real x = std::get<0>(*it);
-    Real y = std::get<1>(*it);
-    Real Δc = value(coeffs, x) - y;
-
-#pragma omp parallel for
-    for (unsigned j = 0; j < coeffs.size(); j++) {
-      dC[j] += Δc * basisFunctions(coeffs.size(), j, x);
-    }
-  }
-
-  return dC;
-}
-
-Vector underSolve(const Data& data, unsigned N) {
-  unsigned M = data.size();
-  Matrix A(M, N);
-  Vector b(M);
-  for (unsigned i = 0; i < M; i++) {
-    Real x = std::get<0>(data[i]);
-    Real y = std::get<1>(data[i]);
-    b[i] = y;
-    for (unsigned j = 0; j < N; j++) {
-      A(i, j) = basisFunctions(N, j, x);
-    }
-  }
-
-  return A.colPivHouseholderQr().solve(b);
-}
-
-Vector stochasticGradientDescent(const Data& data, const Vector& a₀, unsigned nBatches, long unsigned maxSteps, Real ε = 1e-12) {
-  Vector xₜ = a₀;
-  Real Hₜ;
-  Real α = 1.0;
-  Real m;
-  Vector ∇H;
-  long unsigned steps = 0;
-
-  unsigned batchSize = data.size() / nBatches;
-  Data::const_iterator batchBegin = data.begin();
-  Data::const_iterator batchEnd = data.begin() + batchSize;
-  Data::const_iterator effectiveEnd = data.begin() + batchSize * nBatches;
-
-  while (
-    Hₜ = cost(batchBegin, batchEnd, xₜ),
-    ∇H = dCost(batchBegin, batchEnd, xₜ),
-    m = ∇H.squaredNorm(),
-    m > ε && steps < maxSteps
-  ) {
-    Vector xₜ₊₁;
-    Real Hₜ₊₁;
-
-    while (
-      xₜ₊₁ = xₜ - α * ∇H, Hₜ₊₁ = cost(batchBegin, batchEnd, xₜ₊₁),
-      Hₜ₊₁ > Hₜ - 0.25 * α * m
-    ) {
-      α /= 2;
-    }
-
-    xₜ = xₜ₊₁;
-    α *= 1.25;
-    steps++;
-
-    if (batchEnd == data.end()) {
-      batchBegin = data.begin();
-      batchEnd = data.begin() + batchSize;
-    } else if (batchEnd == effectiveEnd) {
-      batchBegin = effectiveEnd;
-      batchEnd = data.end();
-    } else {
-      batchBegin += batchSize;
-      batchEnd += batchSize;
-    }
-  }
-
-  return xₜ;
-}
-
-Data generateData(Real(*f)(Real), unsigned M, Real σ, Rng& r) {
-  Data data;
-  data.reserve(M);
-
-  for (unsigned i = 0; i < M; i++) {
-    Real x = ((Real)i) / (M - 1.0);
-    data.push_back({x, f(x) + r.variate<Real, std::normal_distribution>(0, σ)});
-  }
-
-  return data;
-}
+#include "fits.hpp"
 
 int main(int argc, char* argv[]) {
   unsigned Nend = 1;
diff --git a/fits.hpp b/fits.hpp
new file mode 100644
index 0000000..c6e56fa
--- /dev/null
+++ b/fits.hpp
@@ -0,0 +1,139 @@
+#include "pcg-cpp/include/pcg_random.hpp"
+#include "randutils/randutils.hpp"
+
+#include "eigen/Eigen/Dense"
+
+using Rng = randutils::random_generator<pcg32>;
+
+using Real = double;
+using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
+using Matrix = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
+using Data = std::vector<std::tuple<Real, Real>>;
+
+#ifndef FITS_ABS_BASIS
+inline Real basisFunctions(unsigned N, unsigned i, Real x) {
+  return std::legendre(i, 2.0 * (x - 0.5));
+}
+#endif
+
+#ifdef FITS_ABS_BASIS
+inline Real basisFunctions(unsigned N, unsigned i, Real x) {
+  return abs(x - i / (N - 1.0));
+}
+#endif
+
+Real value(const Vector& coeffs, Real x) {
+  Real v = 0;
+
+  for (unsigned j = 0; j < coeffs.size(); j++) {
+    v += coeffs[j] * basisFunctions(coeffs.size(), j, x);
+  }
+
+  return v;
+}
+
+Real cost(Data::const_iterator batchBegin, Data::const_iterator batchEnd, const Vector& coeffs) {
+  Real c = 0;
+
+#pragma omp parallel for reduction(+:c)
+  for (Data::const_iterator it = batchBegin; it != batchEnd; it++) {
+    Real x = std::get<0>(*it);
+    Real y = std::get<1>(*it);
+    c += 0.5 * pow(value(coeffs, x) - y, 2);
+  }
+
+  return c;
+}
+
+Vector dCost(Data::const_iterator batchBegin, Data::const_iterator batchEnd, const Vector& coeffs) {
+  Vector dC = Vector::Zero(coeffs.size());
+
+  for (Data::const_iterator it = batchBegin; it != batchEnd; it++) {
+    Real x = std::get<0>(*it);
+    Real y = std::get<1>(*it);
+    Real Δc = value(coeffs, x) - y;
+
+#pragma omp parallel for
+    for (unsigned j = 0; j < coeffs.size(); j++) {
+      dC[j] += Δc * basisFunctions(coeffs.size(), j, x);
+    }
+  }
+
+  return dC;
+}
+
+Vector underSolve(const Data& data, unsigned N) {
+  unsigned M = data.size();
+  Matrix A(M, N);
+  Vector b(M);
+  for (unsigned i = 0; i < M; i++) {
+    Real x = std::get<0>(data[i]);
+    Real y = std::get<1>(data[i]);
+    b[i] = y;
+    for (unsigned j = 0; j < N; j++) {
+      A(i, j) = basisFunctions(N, j, x);
+    }
+  }
+
+  return A.colPivHouseholderQr().solve(b);
+}
+
+Vector stochasticGradientDescent(const Data& data, const Vector& a₀, unsigned nBatches, long unsigned maxSteps, Real ε = 1e-12) {
+  Vector xₜ = a₀;
+  Real Hₜ;
+  Real α = 1.0;
+  Real m;
+  Vector ∇H;
+  long unsigned steps = 0;
+
+  unsigned batchSize = data.size() / nBatches;
+  Data::const_iterator batchBegin = data.begin();
+  Data::const_iterator batchEnd = data.begin() + batchSize;
+  Data::const_iterator effectiveEnd = data.begin() + batchSize * nBatches;
+
+  while (
+    Hₜ = cost(batchBegin, batchEnd, xₜ),
+    ∇H = dCost(batchBegin, batchEnd, xₜ),
+    m = ∇H.squaredNorm(),
+    m > ε && steps < maxSteps
+  ) {
+    Vector xₜ₊₁;
+    Real Hₜ₊₁;
+
+    while (
+      xₜ₊₁ = xₜ - α * ∇H, Hₜ₊₁ = cost(batchBegin, batchEnd, xₜ₊₁),
+      Hₜ₊₁ > Hₜ - 0.25 * α * m
+    ) {
+      α /= 2;
+    }
+
+    xₜ = xₜ₊₁;
+    α *= 1.25;
+    steps++;
+
+    if (batchEnd == data.end()) {
+      batchBegin = data.begin();
+      batchEnd = data.begin() + batchSize;
+    } else if (batchEnd == effectiveEnd) {
+      batchBegin = effectiveEnd;
+      batchEnd = data.end();
+    } else {
+      batchBegin += batchSize;
+      batchEnd += batchSize;
+    }
+  }
+
+  return xₜ;
+}
+
+Data generateData(Real(*f)(Real), unsigned M, Real σ, Rng& r) {
+  Data data;
+  data.reserve(M);
+
+  for (unsigned i = 0; i < M; i++) {
+    Real x = ((Real)i) / (M - 1.0);
+    data.push_back({x, f(x) + r.variate<Real, std::normal_distribution>(0, σ)});
+  }
+
+  return data;
+}
diff --git a/fits.nb b/fits.nb
index aa28e7a..052ab44 100644
--- a/fits.nb
+++ b/fits.nb
@@ -10,10 +10,10 @@
 NotebookFileLineBreakTest
 NotebookFileLineBreakTest
 NotebookDataPosition[       154,          7]
-NotebookDataLength[   1530793,      27220]
-NotebookOptionsPosition[   1524362,      27105]
-NotebookOutlinePosition[   1524757,      27121]
-CellTagsIndexPosition[   1524714,      27118]
+NotebookDataLength[   2831857,      49713]
+NotebookOptionsPosition[   2825203,      49596]
+NotebookOutlinePosition[   2825598,      49612]
+CellTagsIndexPosition[   2825555,      49609]
 WindowFrame->Normal*)
 
 (* Beginning of Notebook Content *)
@@ -6996,21 +6996,6 @@ aA8A0XlEJQ==
 Cell[CellGroupData[{
 
 Cell[BoxData[
- RowBox[{"Length", "[", 
-  RowBox[{"as", "[", 
-   RowBox[{"[", 
-    RowBox[{"-", "31"}], "]"}], "]"}], "]"}]], "Input",
- CellChangeTimes->{{3.948274025275671*^9, 3.948274142941441*^9}},
- CellLabel->"In[49]:=",ExpressionUUID->"40855202-d9c2-46f6-ae45-c3c37ee795d0"],
-
-Cell[BoxData["80"], "Output",
- CellChangeTimes->{{3.948274026529709*^9, 3.948274143213243*^9}},
- CellLabel->"Out[49]=",ExpressionUUID->"a961f55f-4340-44b4-8a5e-96fef706b7c0"]
-}, Open  ]],
-
-Cell[CellGroupData[{
-
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  RowBox[{"overparamfit", "=", 
   RowBox[{"Show", "[", 
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@@ -9495,6 +9480,22494 @@ aA8A0XlEJQ==
  CellLabel->"Out[50]=",ExpressionUUID->"eaa9a4dd-4bdb-4a04-9840-d5957524e949"]
 }, Open  ]],
 
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"asExtra", "=", 
+   RowBox[{
+   "Import", 
+    "[", "\"\<~/doc/professional/presentations/2025/ictp-saifr_colloquium/\
+poly_fits_extra.dat\>\"", "]"}]}], ";"}]], "Input",
+ CellChangeTimes->{{3.948282608996006*^9, 3.948282629411549*^9}},
+ CellLabel->
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+
+Cell[CellGroupData[{
+
+Cell[BoxData[
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+  RowBox[{
+   RowBox[{
+    RowBox[{"Show", "[", 
+     RowBox[{
+      RowBox[{"Plot", "[", 
+       RowBox[{
+        RowBox[{"{", 
+         RowBox[{"#", ".", 
+          RowBox[{"polyBasis", "[", 
+           RowBox[{"Length", "[", "#", "]"}], "]"}]}], "}"}], ",", 
+        RowBox[{"{", 
+         RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", 
+        RowBox[{"PlotRange", "->", 
+         RowBox[{"{", 
+          RowBox[{
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+        RowBox[{"ImageSize", "->", "200"}]}], "]"}], ",", 
+      RowBox[{"ListPlot", "[", 
+       RowBox[{"testdata", ",", 
+        RowBox[{"PlotStyle", "->", "Black"}]}], "]"}]}], "]"}], "&"}], "/@", 
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diff --git a/ictp-saifr_colloquium.tex b/ictp-saifr_colloquium.tex
index dc152f8..1fa3a0b 100644
--- a/ictp-saifr_colloquium.tex
+++ b/ictp-saifr_colloquium.tex
@@ -377,11 +377,11 @@
 
       \medskip
 
-      Overparameterized fits are not unique: $M$ constraints
+      Overparameterized fits are not unique: $\chi^2=0$ gives $M$ constraints
       \[
         0=y_i-\hat f(x_i\mid\pmb a)\qquad\text{for all $1\leq i\leq M$}
       \]
-      plus $N$ unknowns $\pmb a=[a_1,\ldots, a_N]$ gives a manifold of $N-M$ dimensions
+      with $N$ unknowns $\pmb a=[a_1,\ldots, a_N]$ gives a manifold of $N-M$ dimensions
 
       \medskip
 
@@ -393,12 +393,47 @@
 
       \bigskip
 
-      \includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly.pdf}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly.pdf}
+        \onslide<2>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_1.pdf}
+        \onslide<3>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_2.pdf}
+        \onslide<4>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_3.pdf}
+        \onslide<5>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_6.pdf}
+        \onslide<6>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_7.pdf}
+        \onslide<7>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_8.pdf}
+        \onslide<8>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_9.pdf}
+      \end{overprint}
     \end{column}
   \end{columns}
 \end{frame}
 
 \begin{frame}
+  \frametitle{Overparamaterized curve fitting and algorithms}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Overparameterized fits found with gradient descent algorithm: take small
+      steps in the direction $\nabla\chi^2$ until $\|\nabla\chi^2\|<\epsilon$
+
+      \medskip
+
+      Result of descent depends on initial condition: what $\pmb a$ to start with?
+
+      \medskip
+
+      \textbf{Unexpected fact:} gradient descent with small initialization equivalent to unique optimum in \emph{regularized} problem:
+      \[
+        \chi^2_\text{eff}(\pmb a\mid\text{data})=\chi^2(\pmb a\mid\text{data})+\lambda\|\pmb a\|^2
+      \]
+    \end{column}
+    \begin{column}{0.5\textwidth}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+\end{frame}
+
+\begin{frame}
   \frametitle{The Euler characteristic \boldmath{$\chi$}}
   \begin{columns}
     \begin{column}{0.5\textwidth}
diff --git a/many_over.cpp b/many_over.cpp
new file mode 100644
index 0000000..5785ad3
--- /dev/null
+++ b/many_over.cpp
@@ -0,0 +1,67 @@
+#include <getopt.h>
+#include <iomanip>
+
+#include "fits.hpp"
+
+int main(int argc, char* argv[]) {
+  unsigned N = 80;
+
+  int opt;
+
+  while ((opt = getopt(argc, argv, "M:s:S:B:i:N:")) != -1) {
+    switch (opt) {
+    case 'N':
+      N = (unsigned)atof(optarg);
+      break;
+    default:
+      exit(1);
+    }
+  }
+
+  std::string firstline;
+  std::getline(std::cin, firstline);
+  std::stringstream ss;
+  ss << firstline;
+
+  Data data;
+
+  while (!ss.eof()) {
+    Real x, y;
+    ss >> x;
+    ss >> y;
+    data.push_back({x,y});
+  }
+
+  Rng r;
+
+  std::cout << std::setprecision(15);
+
+  for (unsigned i = 0; i < 5; i++) {
+    Vector a₀ = Vector::Zero(N);
+    for (Real& aa : a₀) {
+      aa = r.variate<Real, std::normal_distribution>(0, 2e-2);
+    }
+
+    Vector a = stochasticGradientDescent(data, a₀, 10, 1e12);
+
+    for (Real ai : a) {
+      std::cout << ai << " ";
+    }
+    std::cout << std::endl;
+  }
+  for (unsigned i = 0; i < 5; i++) {
+    Vector a₀ = Vector::Zero(N);
+    for (Real& aa : a₀) {
+      aa = r.variate<Real, std::normal_distribution>(0, 5);
+    }
+
+    Vector a = stochasticGradientDescent(data, a₀, 10, 1e12);
+
+    for (Real ai : a) {
+      std::cout << ai << " ";
+    }
+    std::cout << std::endl;
+  }
+
+  return 0;
+}
diff --git a/poly_fits_extra.dat b/poly_fits_extra.dat
new file mode 100644
index 0000000..84f2c67
--- /dev/null
+++ b/poly_fits_extra.dat
@@ -0,0 +1,10 @@
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