Fixing issue in SGOT example and SGOT cost function. (#830)

* correction SGOT cost matrix, added to contributor list, move sgot example to other

* updated graphs

* fix plots

* update example

* fix releases.md

---------

Co-authored-by: Sienna O'Shea <osheasienna@gmail.com>
Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>

Author: 3 people

CITATION.cff

@@ -81,10 +81,17 @@ authors:
   - given-names: David
     family-names: Coeurjolly
     affiliation: CNRS, LIRIS
+  - given-names: Thibaut
+    family-names: Germain
+    affiliation: Ecole Polytechnique
+  - given-names: Sienna
+    family-names: O'Shea
+    affiliation: Ecole Polytechnique
   - given-names: Marco
     family-names: Corneli
     affiliation: Université Côte d'Azur
-  - given-names: Ferdinand Genans
+  - given-names: Ferdinand
+    family-names: Genans
     affiliation: Sorbonne Université, LPSM, CNRS
 identifiers:
   - type: url

CONTRIBUTORS.md

@@ -59,6 +59,8 @@ The contributors to this library are:
 * [Julie Delon](https://judelo.github.io/) (GMM OT)
 * [Samuel Boïté](https://samuelbx.github.io/) (GMM OT)
 * [Nathan Neike](https://github.com/nathanneike) (Sparse EMD solver)
+* [Thibaut Germain](https://thibaut-germain.github.io) (SGOT)
+* Sienna O'Shea (SGOT)
 
 
 ## Acknowledgments

RELEASES.md

@@ -38,6 +38,7 @@ This new release adds support for sparse cost matrices and a new lazy exact OT s
 - Update the geomloss wrapper to the new version and API (PR #826)
 - Fix docstrings for `lowrank_gromov_wasserstein_samples` and `lowrank_sinkhorn` (PR #823)
 - Reorganize all tests per backend (PR #828)
+- Update sgot cost function and example (PR #830)
 
 
 #### Closed issues

examples/plot_sgot.py examples/others/plot_sgot.pyexamples/plot_sgot.py renamed to examples/others/plot_sgot.py

@@ -66,33 +66,35 @@
 theta_0 = np.pi / 4
 
 
+def rotation_matrix(theta):
+    return np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])
+
+
 def generate_data(time, tau, freq, theta):
     t_ = np.sin(2 * np.pi * freq[None, :] * time[:, None]) * np.exp(
         -tau[None, :] * time[:, None]
     )
     t_ = t_.sum(axis=1)
     traj_0 = np.zeros((t_.shape[0], 2))
     traj_0[:, 0] = t_
-    rotation_matrix = np.array(
-        [[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]]
-    )
-    traj_0 = traj_0 @ rotation_matrix.T
+    R_ = rotation_matrix(theta)
+    traj_0 = traj_0 @ R_.T
     return traj_0
 
 
 traj_0 = generate_data(time, tau_0, freq_0, theta_0)
+traj_0_proj = traj_0 @ rotation_matrix(theta_0)[:, 0]
 
 
 # plot the observed signal components and their sum
 plt.figure(figsize=(10, 4))
-plt.plot(time, traj_0, label="base trajectory", linewidth=2)
+plt.plot(time, traj_0_proj, label="projected trajectory", linewidth=2)
 plt.xlabel("time")
 plt.ylabel("amplitude")
 plt.legend()
 plt.title(r"Observed scalar signal along $\vec{e}(\theta)$")
 plt.show()
 
-
 # %%
 # 2. Interpret the signal as coming from a continuous linear dynamical system
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -274,12 +276,8 @@ def augment(traj, window_length=2):
 # Processing Systems, 35, pp.4017-4031.
 
 
-def estimator(X, Y, rank=4):
-    # X: (n_samples, n_features)
-    # Y: (n_samples, n_features)
-
-    # estimate operator
-    cxx = X.T @ X
+def estimator(X, Y, rank=4, eps=1e-8):
+    cxx = X.T @ X + eps * np.eye(X.shape[1])
     U, S, Vt = np.linalg.svd(cxx)
     S_inv = np.divide(1, S, out=np.zeros_like(S), where=S != 0)
     cxx_inv_half = Vt.T @ np.diag(np.sqrt(S_inv)) @ U.T
@@ -416,6 +414,24 @@ def estimator(X, Y, rank=4):
 # spectral atoms, taking into account both the location of eigenvalues and the
 # relative geometry of their eigenspaces.
 
+# %%
+# A wider delay window for the SGOT experiments below
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+#
+# The window of length 4 used above is enough to identify a single reference
+# operator, but the experiments below also probe signals whose two modes
+# nearly coincide in frequency (e.g. :math:`\omega_2'\to\omega_1`). Telling
+# such near-degenerate modes apart requires the delay embedding to span
+# enough time to "see" their differing decay, so we re-embed the reference
+# signal with a longer window before running the sweeps.
+
+sgot_window = 10
+Z = augment(traj_0, sgot_window)
+_, B_0_spec_sgot = estimator(Z[:-1], Z[1:])
+D_0_sgot = np.log(B_0_spec_sgot["eig_val"]) * fs
+L_0_sgot = B_0_spec_sgot["eig_vec_left"]
+R_0_sgot = B_0_spec_sgot["eig_vec_right"]
+
 # %%
 # SGOT distance versus rotation angle
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -434,52 +450,66 @@ def estimator(X, Y, rank=4):
 # this experiment isolates the effect of rotating the underlying one-dimensional
 # subspace in the observation plane.
 
-thetas = np.linspace(0, np.pi / 2, 50)
-lst = []
-for i, theta in enumerate(thetas):
-    traj = generate_data(time, tau_0, freq_0, theta)
-    Z = augment(traj, 4)
-    X = Z[:-1]
-    Y = Z[1:]
-    B, B_spec = estimator(X, Y, rank=4)
-    D, R, L = B_spec["eig_val"], B_spec["eig_vec_right"], B_spec["eig_vec_left"]
-    D = np.log(D) * fs
-    lst.append(sgot_metric(D_0, R_0, L_0, D, R, L, eta=0.01))
-
-plt.figure(figsize=(8, 5))
-plt.plot(thetas, lst)
-plt.xlabel("theta")
-plt.ylabel("SGOT distance")
-plt.title("SGOT distance vs rotation angle")
+thetas = np.linspace(0, np.pi / 2, 51)
+rotation_scores = []
+
+for theta in thetas:
+    Z = augment(generate_data(time, tau_0, freq_0, theta), sgot_window)
+    B, B_spec = estimator(Z[:-1], Z[1:])
+    D = np.log(B_spec["eig_val"]) * fs
+    L = B_spec["eig_vec_left"]
+    R = B_spec["eig_vec_right"]
+    rotation_scores.append(
+        sgot_metric(
+            D_0_sgot, R_0_sgot, L_0_sgot, D, R, L, eta=0.9, grassmann_metric="chordal"
+        )
+    )
+
+fig, ax = plt.subplots(figsize=(7, 4))
+ax.plot(thetas, rotation_scores, linewidth=1.8)
+ax.axvline(theta_0, color="gray", linestyle="--", linewidth=0.8)
+ax.set_xlabel(r"Rotation angle $\theta$ (rad)")
+ax.set_ylabel(r"$d_S$")
+ax.set_title("SGOT distance vs. rotation angle")
+fig.tight_layout()
 plt.show()
 
 # %%
 # Comparison across Grassmannian metrics for SGOT distance versus rotation angle
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-thetas = np.linspace(0, np.pi / 2, 50)
-lst = []
-for i, theta in enumerate(thetas):
-    traj = generate_data(time, tau_0, freq_0, theta)
-    Z = augment(traj, 4)
-    X = Z[:-1]
-    Y = Z[1:]
-    B, B_spec = estimator(X, Y, rank=4)
-    D, R, L = B_spec["eig_val"], B_spec["eig_vec_right"], B_spec["eig_vec_left"]
-    D = np.log(D) * fs
-    lst1 = []
-    for name in ["chordal", "martin", "geodesic", "procrustes"]:
-        lst1.append(sgot_metric(D_0, R_0, L_0, D, R, L, eta=0.9, grassmann_metric=name))
-    lst.append(lst1)
-lst2 = np.array(lst)
-plt.figure(figsize=(8, 5))
-for i, name in enumerate(["chordal", "martin", "geodesic", "procrustes"]):
-    plt.plot(thetas, lst2[:, i], label=name)
-
-plt.xlabel("theta")
-plt.ylabel("SGOT distance")
-plt.title("SGOT distance vs rotation angle")
-plt.legend()
+metrics = ["chordal", "geodesic", "procrustes", "martin"]
+styles = {"chordal": "-", "geodesic": "--", "procrustes": "-.", "martin": ":"}
+rotation_results = {m: [] for m in metrics}
+
+for theta in thetas:
+    Z = augment(generate_data(time, tau_0, freq_0, theta), sgot_window)
+    B, B_spec = estimator(Z[:-1], Z[1:])
+    D = np.log(B_spec["eig_val"]) * fs
+    L = B_spec["eig_vec_left"]
+    R = B_spec["eig_vec_right"]
+    for m in metrics:
+        rotation_results[m].append(
+            sgot_metric(
+                D_0_sgot, R_0_sgot, L_0_sgot, D, R, L, eta=0.9, grassmann_metric=m
+            )
+        )
+
+fig, ax = plt.subplots(figsize=(7, 4))
+for m in metrics:
+    ax.plot(thetas, rotation_results[m], styles[m], label=m, linewidth=1.8)
+ax.axvline(
+    theta_0,
+    color="gray",
+    linestyle="--",
+    linewidth=0.8,
+    label=r"$\theta_0 = \pi/4$ (reference)",
+)
+ax.set_xlabel(r"Rotation angle $\theta$ (rad)")
+ax.set_ylabel(r"$d_S$")
+ax.set_title("SGOT distance vs. rotation angle across Grassmannian metrics")
+ax.legend()
+fig.tight_layout()
 plt.show()
 
 # %%
@@ -501,38 +531,38 @@ def estimator(X, Y, rank=4):
 # distance changes as a function of the perturbed frequency :math:`\omega_2'`.
 
 omegas = np.linspace(0.5, 3.0, 21)
-methods = ["chordal", "martin", "geodesic", "procrustes"]
-scores_omega = []
-theta = theta_0
+frequency_scores = {m: [] for m in metrics}
 
-eta_fixed = 0.9
 for omega in omegas:
-    freq_1 = np.array([freq_0[0], omega])
-    traj = generate_data(time, tau_0, freq_1, theta)
-    Z = augment(traj, 4)
-    X = Z[:-1]
-    Y = Z[1:]
-
-    B, B_spec = estimator(X, Y, rank=4)
-    D, R, L = B_spec["eig_val"], B_spec["eig_vec_right"], B_spec["eig_vec_left"]
-    D = np.log(D) * fs
-
-    row = []
-    for name in methods:
-        row.append(
-            sgot_metric(D_0, R_0, L_0, D, R, L, eta=eta_fixed, grassmann_metric=name)
+    Z = augment(
+        generate_data(time, tau_0, np.array([freq_0[0], omega]), theta_0), sgot_window
+    )
+    B, B_spec = estimator(Z[:-1], Z[1:])
+    D = np.log(B_spec["eig_val"]) * fs
+    L = B_spec["eig_vec_left"]
+    R = B_spec["eig_vec_right"]
+    for m in metrics:
+        frequency_scores[m].append(
+            sgot_metric(
+                D_0_sgot, R_0_sgot, L_0_sgot, D, R, L, eta=0.9, grassmann_metric=m
+            )
         )
-    scores_omega.append(row)
-
-scores_omega = np.array(scores_omega)
-plt.figure(figsize=(8, 5))
-for i, name in enumerate(methods):
-    plt.plot(omegas, scores_omega[:, i], label=name)
 
-plt.xlabel("omega")
-plt.ylabel("SGOT distance")
-plt.title("SGOT distance vs omega")
-plt.legend()
+fig, ax = plt.subplots(figsize=(7, 4))
+for m in metrics:
+    ax.plot(omegas, frequency_scores[m], styles[m], label=m, linewidth=1.8)
+ax.axvline(
+    freq_0[1],
+    color="gray",
+    linestyle="--",
+    linewidth=0.8,
+    label=r"$\omega_2 = 2.0$ Hz (reference)",
+)
+ax.set_xlabel(r"Frequency $\omega_2'$ (Hz)")
+ax.set_ylabel(r"$d_S$")
+ax.set_title("SGOT distance vs. frequency across Grassmannian metrics")
+ax.legend()
+fig.tight_layout()
 plt.show()
 
 # %%
@@ -553,47 +583,28 @@ def estimator(X, Y, rank=4):
 # In this way, both modes share the same modified decay parameter
 # :math:`\tau`, allowing us to isolate the influence of dissipation on the SGOT
 # distance.
-decays = np.linspace(0.1, 3.0, 20)  # adjust range as needed
-methods = ["chordal", "martin", "geodesic", "procrustes"]
-scores_decay = []
-theta = theta_0
-
-for tau in decays:
-    freq_1 = np.array([freq_0[0], recovered_freqs[1]])
-    tau_1 = np.array([tau, tau])  # or whatever structure your generator expects
-
-    traj = generate_data(time, tau_1, freq_1, theta)
-    Z = augment(traj, 4)
-    X = Z[:-1]
-    Y = Z[1:]
-
-    B, B_spec = estimator(X, Y, rank=4)
-    D, R, L = B_spec["eig_val"], B_spec["eig_vec_right"], B_spec["eig_vec_left"]
-    D = np.log(D) * fs
-
-    row = []
-    for name in methods:
-        row.append(
+taus = np.linspace(0.1, 3.0, 21)
+decay_scores = {m: [] for m in metrics}
+
+for tau in taus:
+    Z = augment(generate_data(time, np.array([tau, tau]), freq_0, theta_0), sgot_window)
+    B, B_spec = estimator(Z[:-1], Z[1:])
+    D = np.log(B_spec["eig_val"]) * fs
+    L = B_spec["eig_vec_left"]
+    R = B_spec["eig_vec_right"]
+    for m in metrics:
+        decay_scores[m].append(
             sgot_metric(
-                D_0,
-                R_0,
-                L_0,
-                D,
-                R,
-                L,
-                eta=0.9,  # keep eta fixed here
-                grassmann_metric=name,
+                D_0_sgot, R_0_sgot, L_0_sgot, D, R, L, eta=0.9, grassmann_metric=m
             )
         )
-    scores_decay.append(row)
 
-scores_decay = np.array(scores_decay)
-plt.figure(figsize=(8, 5))
-for i, name in enumerate(methods):
-    plt.plot(decays, scores_decay[:, i], label=name)
-
-plt.xlabel("decay")
-plt.ylabel("SGOT distance")
-plt.title("SGOT distance vs decay")
-plt.legend()
+fig, ax = plt.subplots(figsize=(7, 4))
+for m in metrics:
+    ax.plot(taus, decay_scores[m], styles[m], label=m, linewidth=1.8)
+ax.set_xlabel(r"Decay rate $\tau$")
+ax.set_ylabel(r"$d_S$")
+ax.set_title("SGOT distance vs. decay across Grassmannian metrics")
+ax.legend()
+fig.tight_layout()
 plt.show()

ot/sgot.py

@@ -124,7 +124,7 @@ def _delta_matrix_1d(Rs, Ls, Rt, Lt, nx=None, eps=1e-12):
     Ltn = _normalize_columns(Lt, nx=nx, eps=eps)
 
     Cr = nx.dot(nx.conj(Rsn).T, Rtn)
-    Cl = nx.dot(nx.conj(Lsn).T, Ltn)
+    Cl = nx.dot(Lsn.T, nx.conj(Ltn))
 
     delta = nx.abs(Cr * Cl)
     delta = nx.clip(delta, 0.0, 1.0)