Add reproducible code for matrix exponential simulations (#37)

* Add matrix exponential simulations

* Fix bug enforcing real eigen values

* Add label to plots

* Minor plot change

* Remove import

* Add figures

* Add comments to run

* Better casing

examples/matrix_exponentials/.gitignore

@@ -0,0 +1 @@
+*.pkl

examples/matrix_exponentials/matrix_generation.py

@@ -0,0 +1,13 @@
+from jax import random, numpy as jnp
+from scipy.stats import ortho_group
+
+
+def wishart(d: int, key: random.PRNGKey) -> jnp.ndarray:
+    n = 2 * d  # degrees of freedom
+    G = random.normal(key, shape=(d, n))
+    A_wishart = (G @ G.T) / n
+    return A_wishart
+
+
+def orthogonal(d: int, _) -> jnp.ndarray:
+    return ortho_group.rvs(d)

examples/matrix_exponentials/plot.py

@@ -0,0 +1,138 @@
+import pickle
+import numpy as np
+import matplotlib.pyplot as plt
+import argparse
+
+
+parser = argparse.ArgumentParser()
+parser.add_argument("--save_dir", type=str)
+args = parser.parse_args()
+
+
+matrix_type = args.save_dir.split("/")[-1].split("_")[1].split(".")[0]
+
+
+# use latex for plots
+plt.rc("text", usetex=True)
+# set font
+plt.rc("font", family="serif")
+# set font size
+plt.rcParams.update({"font.size": 10})
+
+colors = [
+    plt.cm.viridis(0.2),
+    plt.cm.viridis(0.4),
+    plt.cm.viridis(0.6),
+    plt.cm.viridis(0.8),
+]
+
+
+results = pickle.load(open(args.save_dir, "rb"))
+dt = results["dt"]
+NT = results["err_abs"].shape[-1]
+D = results["D"]
+e0_abs = 8.0 if matrix_type == "wishart" else 19.0
+ylabel_abs = (
+    r"$|| \bar{C} - \exp(-A)||_F$"
+    if matrix_type == "wishart"
+    else r"$|| \bar{C} - \exp(-M)||_F$"
+)
+e0_rel = 0.9
+ylabel_rel = (
+    r"$\frac{|| \bar{C} - \exp(-A)||_F}{||\exp(-A)||_F}$"
+    if matrix_type == "wishart"
+    else r"$\frac{|| \bar{C} - \exp(-M)||_F}{||\exp(-M)||_F}$"
+)
+fig_label = "(A)" if matrix_type == "wishart" else "(B)"
+
+
+def plot(err, ylabel, e0, save_path, d=False, d_squared=False, fig_label=None):
+    T = np.arange(NT) * dt
+    err_mean = err.mean(axis=0)
+
+    # find time where error crosses threshold
+    TC = np.zeros(len(D))
+    for i in range(len(D)):
+        TC[i] = np.min(T[10:][err_mean[i, 10:] < e0])
+
+    plt.figure(figsize=(7, 4.5))
+
+    if fig_label is not None:
+        plt.gcf().text(0.02, 0.93, fig_label, fontsize=22)
+
+    for i in range(len(D)):
+        plt.plot(T, err_mean[i], color=colors[i])
+
+    # Add error bars
+    for i in range(len(D)):
+        plt.fill_between(
+            T,
+            err_mean[i] - err[:, i].std(axis=0),
+            err_mean[i] + err[:, i].std(axis=0),
+            color=colors[i],
+            alpha=0.3,
+            zorder=0,
+        )
+
+    plt.loglog()
+    plt.legend(["d = " + str(D[i]) for i in range(len(D))], loc="upper right")
+    plt.xlabel(r"Time ($\mu$s)", fontsize=18)
+    plt.ylabel(ylabel, fontsize=18)
+
+    # show threshold as horizontal line
+    plt.axhline(e0, color="k", linestyle="--")
+    # show crossing times as vertical lines
+    for i in range(len(D)):
+        plt.axvline(TC[i], color=colors[i], linestyle="--")
+
+    plt.xlim(30, T[-1])
+
+    # inset plot showing crossing time as a function of dimension
+    ax = plt.axes([0.17, 0.22, 0.3, 0.35])
+    ax.tick_params(axis="y", direction="in", pad=-22)
+    ax.tick_params(axis="x", direction="in", pad=-15)
+
+    for i in range(len(D)):
+        ax.scatter(D[i], TC[i], color=colors[i], zorder=10)
+
+    ts = np.array([10, 2000])
+
+    if d:
+        plt.plot(ts, 100 * ts, color="black", linestyle="--")
+        plt.text(600, 8e4, s=r"$t_C = d$", rotation=25)
+
+    if d_squared:
+        plt.plot(ts, 0.3 * ts**2, color="black", linestyle="--")
+        plt.text(550, 1.7e5, s=r"$t_C = d^2$", rotation=25)
+
+    plt.plot(D, TC, color="black", zorder=0)
+    plt.xlim(20, 1500)
+
+    plt.loglog()
+    plt.xlabel(r"$d$", fontsize=15)
+    plt.ylabel(r"$t_C$", fontsize=15)
+    plt.minorticks_off()
+
+    plt.tight_layout()
+    plt.savefig(save_path, dpi=300)
+    plt.show()
+
+
+plot(
+    results["err_abs"],
+    ylabel_abs,
+    e0_abs,
+    f"examples/matrix_exponentials/{matrix_type}_abs.pdf",
+    d_squared=True,
+    fig_label=fig_label,
+)
+
+
+plot(
+    results["err_rel"],
+    ylabel_rel,
+    e0_rel,
+    f"examples/matrix_exponentials/{matrix_type}_rel.pdf",
+    d=True,
+    fig_label=fig_label,
+)

examples/matrix_exponentials/run.py

@@ -0,0 +1,114 @@
+from jax import random, jit, config, numpy as jnp
+from jax.scipy.linalg import expm
+from jax.lax import scan
+import numpy as np
+import argparse
+from tqdm import tqdm
+import thermox
+import pickle
+
+from examples.matrix_exponentials import matrix_generation
+
+# Set the precision of the computation
+config.update("jax_enable_x64", True)
+
+# Set seed for orthogonal matrix generation
+np.random.seed(42)
+
+# Load n_repeats, matrix_type and alpha from the command line
+parser = argparse.ArgumentParser()
+parser.add_argument("--n_repeats", type=int, default=1)
+parser.add_argument("--matrix_type", type=str, default="wishart")
+parser.add_argument("--alpha", type=float, default=0.0)
+args = parser.parse_args()
+get_matrix = getattr(matrix_generation, args.matrix_type)
+alpha = args.alpha
+
+# Jit for speed (avoid recompilation)
+sample = jit(thermox.sample)
+
+# Hyperparameters shared across all experiments
+NT = 10000
+dt = 12
+ts = jnp.arange(NT) * dt
+N_burn = 0
+keys = random.split(random.PRNGKey(42), args.n_repeats)
+gamma = 1
+beta = 1
+D = [64, 128, 256, 512]
+
+
+# Function to compute array of autocovariance errors from samples
+@jit
+def samps_to_autocovs_errs(samps, true_exp):
+    def body_func(prev_mat, n):
+        new_mat = prev_mat * n / (n + 1) + jnp.outer(samps[n], samps[n - 1]) / (n + 1)
+        err = jnp.linalg.norm(new_mat * jnp.exp(alpha) - true_exp)
+        return new_mat, err
+
+    return scan(
+        body_func,
+        jnp.zeros((samps.shape[1], samps.shape[1])),
+        jnp.arange(1, samps.shape[0]),
+    )[1]
+
+
+# Initialize arrays to store errors
+err_abs = np.zeros((args.n_repeats, len(D), NT))
+err_rel = np.zeros_like(err_abs)
+
+# Loop over repeats and dimensions
+for repeat in tqdm(range(args.n_repeats)):
+    key = keys[repeat]
+    for i in range(len(D)):
+        d = D[i]
+        print(f"Repeat {repeat}/{args.n_repeats}, \t D = {d}")
+
+        A = get_matrix(d, key)
+        exact_exp_min_A = expm(-A)
+
+        # Shift and scale A and compute symmetrized B
+        A_shifted = (A + alpha * jnp.eye(A.shape[0])) / dt
+        B = A_shifted + A_shifted.T
+
+        # Print eigenvalues
+        A_shifted_lambda_min = jnp.min(jnp.linalg.eig(A_shifted / gamma)[0].real)
+        print("A Eig min: ", A_shifted_lambda_min)
+
+        D_lambda_min = jnp.min(jnp.linalg.eig(B / (gamma * beta))[0].real)
+        print("D Eig min: ", D_lambda_min)
+
+        # Initialize at zeros
+        x0 = np.zeros(d)
+
+        # Run the sampler
+        X = sample(
+            key,
+            ts,
+            x0,
+            A_shifted / gamma,
+            np.zeros(d),
+            B / (gamma * beta),
+        )
+
+        # Compute absolute error
+        err_abs = samps_to_autocovs_errs(X, exact_exp_min_A)
+        err_abs[repeat, i, 1:] = err_abs
+
+        # Compute relative error
+        err_rel[repeat, i, 1:] = err_abs / jnp.linalg.norm(exact_exp_min_A)
+
+    # Save results (overwrites after each repeat)
+    with open(
+        f"examples/matrix_exponentials/results_{args.matrix_type}.pkl", "wb"
+    ) as f:
+        pickle.dump(
+            {
+                "D": D,
+                "dt": dt,
+                "alpha": alpha,
+                "err_abs": err_abs,
+                "err_rel": err_rel,
+            },
+            f,
+        )

thermox/utils.py

@@ -28,18 +28,14 @@ def preprocess_drift_matrix(A: Array) -> ProcessedDriftMatrix:
     """
 
     A_eigvals, A_eigvecs = eig(A + 0.0j)
-
-    A_eigvals = A_eigvals.real
-    A_eigvecs = A_eigvecs.real
-
     A_eigvecs_inv = jnp.linalg.inv(A_eigvecs)
 
     symA = 0.5 * (A + A.T)
     symA_eigvals, symA_eigvecs = jnp.linalg.eigh(symA)
 
     return ProcessedDriftMatrix(
         A,
-        A_eigvals.real,
+        A_eigvals,
         A_eigvecs,
         A_eigvecs_inv,
         symA_eigvals,