# SKA Time-Invariance Explorer - Gradio App
import torch
import torch.nn as nn
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from torchvision import datasets, transforms
import gradio as gr
# Load MNIST from local data
transform = transforms.Compose([transforms.ToTensor()])
mnist_dataset = datasets.MNIST(root='./data', train=True, download=False, transform=transform)
# Fixed architecture and characteristic time as per arXiv:2504.03214v1
LAYER_SIZES = [256, 128, 64, 10]
TAU = 0.5
# Exact 6 (eta, K) configurations from the paper — all satisfy eta * K = 0.5
CONFIGS = [
(0.020, 25),
(0.010, 50),
(0.005, 100),
(0.0033, 150),
(0.0025, 200),
(0.001, 500),
]
CONFIG_COLORS = ['#1F77B4', '#FF7F0E', '#2CA02C', '#D62728', '#9467BD', '#8C564B']
class SKAModel(nn.Module):
def __init__(self, input_size=784, layer_sizes=[256, 128, 64, 10], K=50):
super(SKAModel, self).__init__()
self.input_size = input_size
self.layer_sizes = layer_sizes
self.K = K
self.weights = nn.ParameterList()
self.biases = nn.ParameterList()
prev_size = input_size
for size in layer_sizes:
self.weights.append(nn.Parameter(torch.randn(prev_size, size) * 0.01))
self.biases.append(nn.Parameter(torch.zeros(size)))
prev_size = size
self.Z = [None] * len(layer_sizes)
self.Z_prev = [None] * len(layer_sizes)
self.D = [None] * len(layer_sizes)
self.D_prev = [None] * len(layer_sizes)
self.delta_D = [None] * len(layer_sizes)
self.entropy = [None] * len(layer_sizes)
self.entropy_history = [[] for _ in range(len(layer_sizes))]
self.cosine_history = [[] for _ in range(len(layer_sizes))]
self.output_history = []
def forward(self, x):
batch_size = x.shape[0]
x = x.view(batch_size, -1)
for l in range(len(self.layer_sizes)):
z = torch.mm(x, self.weights[l]) + self.biases[l]
d = torch.sigmoid(z)
self.Z[l] = z
self.D[l] = d
x = d
return x
def calculate_entropy(self):
for l in range(len(self.layer_sizes)):
if self.Z[l] is not None and self.D_prev[l] is not None and self.D[l] is not None and self.Z_prev[l] is not None:
self.delta_D[l] = self.D[l] - self.D_prev[l]
H_lk = (-1 / np.log(2)) * (self.Z[l] * self.delta_D[l])
layer_entropy = torch.sum(H_lk)
self.entropy[l] = layer_entropy.item()
self.entropy_history[l].append(layer_entropy.item())
dot_product = torch.sum(self.Z[l] * self.delta_D[l])
z_norm = torch.norm(self.Z[l])
delta_d_norm = torch.norm(self.delta_D[l])
if z_norm > 0 and delta_d_norm > 0:
cos_theta = dot_product / (z_norm * delta_d_norm)
self.cosine_history[l].append(cos_theta.item())
else:
self.cosine_history[l].append(0.0)
def ska_update(self, inputs, learning_rate=0.01):
for l in range(len(self.layer_sizes)):
if self.delta_D[l] is not None:
prev_output = inputs.view(inputs.shape[0], -1) if l == 0 else self.D_prev[l-1]
d_prime = self.D[l] * (1 - self.D[l])
gradient = -1 / np.log(2) * (self.Z[l] * d_prime + self.delta_D[l])
dW = torch.matmul(prev_output.t(), gradient) / prev_output.shape[0]
self.weights[l] = self.weights[l] - learning_rate * dW
self.biases[l] = self.biases[l] - learning_rate * gradient.mean(dim=0)
def initialize_tensors(self, batch_size):
for l in range(len(self.layer_sizes)):
self.Z[l] = None
self.Z_prev[l] = None
self.D[l] = None
self.D_prev[l] = None
self.delta_D[l] = None
self.entropy[l] = None
self.entropy_history[l] = []
self.cosine_history[l] = []
self.output_history = []
def get_mnist_subset(samples_per_class, data_seed=0):
"""Select N samples per class from MNIST."""
images_list = []
targets = mnist_dataset.targets.numpy()
rng = np.random.RandomState(data_seed)
for digit in range(10):
all_indices = np.where(targets == digit)[0]
rng.shuffle(all_indices)
indices = all_indices[:samples_per_class]
for idx in indices:
img, label = mnist_dataset[idx]
images_list.append(img)
images = torch.stack(images_list)
return images
def run_time_invariance(samples_per_class, data_seed):
samples_per_class = int(samples_per_class)
data_seed = int(data_seed)
inputs = get_mnist_subset(samples_per_class, data_seed)
results = []
for eta, K in CONFIGS:
torch.manual_seed(42)
np.random.seed(42)
model = SKAModel(input_size=784, layer_sizes=LAYER_SIZES, K=K)
model.initialize_tensors(inputs.size(0))
for k in range(K):
model.forward(inputs)
if k > 0:
model.calculate_entropy()
model.ska_update(inputs, eta)
model.D_prev = [d.clone().detach() if d is not None else None for d in model.D]
model.Z_prev = [z.clone().detach() if z is not None else None for z in model.Z]
results.append((eta, K, model.entropy_history, model.cosine_history))
layer_colors = ['#1F77B4', '#FF7F0E', '#2CA02C', '#D62728']
layer_labels = ['Layer 1', 'Layer 2', 'Layer 3', 'Layer 4']
# Plot 1: Entropy — 2x3 grid, one subplot per (eta, K) config, 4 layer curves each
fig1, axes1 = plt.subplots(3, 2, figsize=(14, 18))
for idx, (eta, K, entropy_history, _) in enumerate(results):
ax = axes1[idx // 2][idx % 2]
for l in range(len(LAYER_SIZES)):
ax.plot(entropy_history[l], color=layer_colors[l],
label=layer_labels[l], linewidth=1.5)
ax.set_title(f"Entropy Evolution Across Layers (Single Pass)\nη={eta:.4f}, K={K}", fontsize=10)
ax.set_xlabel("Step Index K")
ax.set_ylabel("Entropy")
ax.legend(fontsize=8)
ax.grid(True)
fig1.suptitle(
f"Time-Invariance — Entropy | T = η·K = {TAU} | [256, 128, 64, 10]",
fontsize=13, y=1.01
)
fig1.tight_layout()
# Plot 2: Cosine alignment — 2x3 grid, one subplot per (eta, K) config, 4 layer curves each
fig2, axes2 = plt.subplots(3, 2, figsize=(14, 18))
for idx, (eta, K, _, cosine_history) in enumerate(results):
ax = axes2[idx // 2][idx % 2]
for l in range(len(LAYER_SIZES)):
ax.plot(cosine_history[l], color=layer_colors[l],
label=layer_labels[l], linewidth=1.5)
ax.set_title(f"Cos(θ) Alignment Evolution Across Layers (Single Pass)\nη={eta:.4f}, K={K}", fontsize=10)
ax.set_xlabel("Step Index K")
ax.set_ylabel("Cos(θ)")
ax.legend(fontsize=8)
ax.grid(True)
fig2.suptitle(
f"Time-Invariance — Cosine Alignment | T = η·K = {TAU} | [256, 128, 64, 10]",
fontsize=13, y=1.01
)
fig2.tight_layout()
return fig1, fig2
with gr.Blocks(title="SKA Time-Invariance Explorer") as demo:
gr.Image("logo.png", show_label=False, height=100, container=False)
gr.Markdown("# SKA Time-Invariance Explorer")
gr.Markdown("Fix the characteristic time T = η · K = 0.5 and run 6 different (η, K) pairs automatically. All entropy and cosine curves collapse onto the same trajectory — revealing the intrinsic timescale of the architecture [256, 128, 64, 10] on MNIST.")
with gr.Row():
with gr.Column(scale=1):
gr.Markdown("**Architecture (fixed):** [256, 128, 64, 10]")
gr.Markdown("**Characteristic time (fixed):** T = η · K = 0.5")
samples_slider = gr.Slider(1, 100, value=100, step=1, label="Samples per class")
seed_slider = gr.Slider(0, 99, value=0, step=1, label="Data seed (shuffle samples)")
run_btn = gr.Button("Run Time-Invariance Test", variant="primary")
gr.Markdown("---")
gr.Markdown("### The 6 configurations")
gr.Markdown(
"| η | K |\n|---|---|\n"
"| 0.0200 | 25 |\n"
"| 0.0100 | 50 |\n"
"| 0.0050 | 100 |\n"
"| 0.0033 | 150 |\n"
"| 0.0025 | 200 |\n"
"| 0.0010 | 500 |"
)
gr.Markdown("---")
gr.Markdown("### Reference Paper")
gr.HTML('arXiv:2504.03214v1')
gr.Markdown("""
**Abstract**
This paper aims to extend the Structured Knowledge Accumulation (SKA) framework recently proposed by mahi. We introduce two core concepts: the Tensor Net function and the characteristic time property of neural learning. First, we reinterpret the learning rate as a time step in a continuous system. This transforms neural learning from discrete optimization into continuous-time evolution. We show that learning dynamics remain consistent when the product of learning rate and iteration steps stays constant. This reveals a time-invariant behavior and identifies an intrinsic timescale of the network. Second, we define the Tensor Net function as a measure that captures the relationship between decision probabilities, entropy gradients, and knowledge change. Additionally, we define its zero-crossing as the equilibrium state between decision probabilities and entropy gradients. We show that the convergence of entropy and knowledge flow provides a natural stopping condition, replacing arbitrary thresholds with an information-theoretic criterion. We also establish that SKA dynamics satisfy a variational principle based on the Euler-Lagrange equation. These findings extend SKA into a continuous and self-organizing learning model. The framework links computational learning with physical systems that evolve by natural laws. By understanding learning as a time-based process, we open new directions for building efficient, robust, and biologically-inspired AI systems.
""")
gr.Markdown("---")
gr.Markdown("### SKA Explorer Suite")
gr.HTML('⬅ All Apps')
gr.Markdown("---")
gr.Markdown("### About this App")
gr.Markdown("Six (η, K) pairs all share the same characteristic time T = η · K = 0.5, the intrinsic timescale of the architecture [256, 128, 64, 10]. Each configuration is run independently and plotted as a function of the step index K. The trajectory shapes remain identical across all configurations while the amplitude scales with η — demonstrating that T is the true timescale of learning, not η or K individually. The characteristic time is the necessary time exposure of the sample to the learning system to complete. T = 0.5 is the characteristic time of the architecture [256, 128, 64, 10] on MNIST.")
with gr.Column(scale=2):
plot_entropy = gr.Plot(label="Entropy — 4 Layers")
plot_cosine = gr.Plot(label="Cosine Alignment — 4 Layers")
run_btn.click(
fn=run_time_invariance,
inputs=[samples_slider, seed_slider],
outputs=[plot_entropy, plot_cosine],
)
demo.launch(server_name="0.0.0.0", server_port=7860, share=True)