{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Differentiable TT Cross-approximation: Example"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import tntorch as tn\n",
"import torch\n",
"torch.set_default_dtype(torch.float64) # Note: float32 is sometimes unstable\n",
"torch.manual_seed(0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example 1: Single Optimization\n",
"\n",
"We will optimize the problem $\\arg \\min_{\\mathcal{T}} \\|\\cos \\mathcal{T}\\|$ (element-wise cosine) using Adam on the adaptive TT-cross interpolation formula."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0000: loss=133.653\n",
"0200: loss=41.3387\n",
"0400: loss=4.57952\n",
"0600: loss=1.38465\n",
"0800: loss=0.503446\n",
"1000: loss=0.270926\n",
"1200: loss=0.19317\n",
"1400: loss=0.153916\n",
"1600: loss=0.130659\n",
"1800: loss=0.115443\n"
]
}
],
"source": [
"# Initial solution; we constrain our search to rank-5 tensors\n",
"t = tn.rand(32, 32, 32, ranks_tt=5, requires_grad=True)\n",
"\n",
"function = lambda x: torch.cos(x)\n",
"\n",
"# Optimization\n",
"optimizer = torch.optim.Adam(t.cores)\n",
"for it in range(2000):\n",
" optimizer.zero_grad()\n",
" \n",
" if it % 400 == 0: # Update cross indices\n",
" _, info = tn.cross(tensors=[t], function=function, verbose=False, return_info=True)\n",
"\n",
" # Forward pass\n",
" t2 = tn.cross_forward(tensors=[t], function=function, info=info)\n",
" loss = tn.norm(t2)\n",
"\n",
" if it % 200 == 0:\n",
" print('{:04}: loss={:g}'.format(it, loss))\n",
" loss.backward()\n",
" optimizer.step()"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"3D TT tensor:\n",
"\n",
" 32 32 32\n",
" | | |\n",
" (0) (1) (2)\n",
" / \\ / \\ / \\\n",
"1 5 5 1\n",
"\n"
]
}
],
"source": [
"print(t)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"tensor([[[1.5708, 1.5707, 1.5707, ..., 1.5705, 4.7115, 1.5705],\n",
" [1.5707, 1.5706, 1.5706, ..., 1.5702, 4.7118, 1.5707],\n",
" [1.5704, 1.5704, 1.5704, ..., 1.5705, 4.7118, 1.5704],\n",
" ...,\n",
" [1.5710, 1.5709, 1.5707, ..., 1.5700, 4.7114, 1.5707],\n",
" [1.5707, 1.5706, 1.5707, ..., 1.5707, 4.7118, 1.5705],\n",
" [1.5702, 1.5702, 1.5702, ..., 1.5706, 4.7118, 1.5703]],\n",
"\n",
" [[1.5703, 1.5703, 1.5704, ..., 1.5704, 4.7117, 1.5704],\n",
" [1.5703, 1.5703, 1.5703, ..., 1.5704, 4.7116, 1.5704],\n",
" [1.5704, 1.5703, 1.5704, ..., 1.5704, 4.7117, 1.5704],\n",
" ...,\n",
" [1.5703, 1.5703, 1.5703, ..., 1.5704, 4.7116, 1.5704],\n",
" [1.5703, 1.5703, 1.5704, ..., 1.5704, 4.7119, 1.5704],\n",
" [1.5704, 1.5704, 1.5704, ..., 1.5704, 4.7115, 1.5704]],\n",
"\n",
" [[1.5702, 1.5703, 1.5703, ..., 1.5704, 4.7114, 1.5703],\n",
" [1.5705, 1.5704, 1.5704, ..., 1.5704, 4.7116, 1.5705],\n",
" [1.5707, 1.5705, 1.5706, ..., 1.5702, 4.7117, 1.5706],\n",
" ...,\n",
" [1.5700, 1.5701, 1.5702, ..., 1.5706, 4.7117, 1.5702],\n",
" [1.5703, 1.5703, 1.5703, ..., 1.5704, 4.7115, 1.5704],\n",
" [1.5707, 1.5706, 1.5705, ..., 1.5703, 4.7114, 1.5705]],\n",
"\n",
" ...,\n",
"\n",
" [[1.5704, 1.5704, 1.5704, ..., 1.5704, 4.7113, 1.5703],\n",
" [1.5706, 1.5705, 1.5705, ..., 1.5703, 4.7116, 1.5706],\n",
" [1.5707, 1.5705, 1.5706, ..., 1.5703, 4.7117, 1.5706],\n",
" ...,\n",
" [1.5703, 1.5703, 1.5703, ..., 1.5704, 4.7116, 1.5704],\n",
" [1.5704, 1.5704, 1.5704, ..., 1.5704, 4.7114, 1.5704],\n",
" [1.5705, 1.5705, 1.5705, ..., 1.5703, 4.7114, 1.5704]],\n",
"\n",
" [[1.5704, 1.5704, 1.5704, ..., 1.5703, 4.7115, 1.5704],\n",
" [1.5704, 1.5705, 1.5704, ..., 1.5706, 4.7112, 1.5703],\n",
" [1.5704, 1.5705, 1.5704, ..., 1.5704, 4.7111, 1.5703],\n",
" ...,\n",
" [1.5702, 1.5703, 1.5704, ..., 1.5707, 4.7115, 1.5703],\n",
" [1.5704, 1.5705, 1.5703, ..., 1.5702, 4.7110, 1.5703],\n",
" [1.5705, 1.5706, 1.5705, ..., 1.5704, 4.7115, 1.5705]],\n",
"\n",
" [[1.5701, 1.5702, 1.5702, ..., 1.5704, 4.7115, 1.5703],\n",
" [1.5704, 1.5704, 1.5704, ..., 1.5704, 4.7116, 1.5704],\n",
" [1.5707, 1.5706, 1.5706, ..., 1.5702, 4.7117, 1.5706],\n",
" ...,\n",
" [1.5699, 1.5700, 1.5701, ..., 1.5706, 4.7118, 1.5702],\n",
" [1.5702, 1.5703, 1.5703, ..., 1.5703, 4.7116, 1.5704],\n",
" [1.5707, 1.5707, 1.5706, ..., 1.5702, 4.7114, 1.5706]]],\n",
" grad_fn=<AsStridedBackward>)\n"
]
}
],
"source": [
"print(t.torch())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example 2: Joint Optimization\n",
"\n",
"We will now find $\\mathcal{T}_1, \\mathcal{T}_2$ such that $\\|\\mathcal{T}_1^2 + \\mathcal{T}_2^2 - 1\\|$ is minimal."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0000: loss=2407.16\n",
"0200: loss=275.716\n",
"0400: loss=125.413\n",
"0600: loss=71.8586\n",
"0800: loss=46.5382\n",
"1000: loss=30.6674\n",
"1200: loss=19.8756\n",
"1400: loss=13.2586\n",
"1600: loss=9.2578\n",
"1800: loss=6.68817\n"
]
}
],
"source": [
"t1 = tn.rand(16, 16, 16, 16, ranks_tt=3, requires_grad=True)\n",
"t2 = tn.rand(16, 16, 16, 16, ranks_tt=3, requires_grad=True)\n",
"\n",
"function = lambda x, y: x**2 + y**2\n",
"\n",
"# Optimization\n",
"optimizer = torch.optim.Adam(t1.cores + t2.cores)\n",
"for it in range(2000):\n",
" optimizer.zero_grad()\n",
" \n",
" if it % 400 == 0: # Update cross indices\n",
" _, info = tn.cross(tensors=[t1, t2], function=function, verbose=False, return_info=True)\n",
"\n",
" # Forward pass\n",
" t3 = tn.cross_forward(tensors=[t1, t2], function=function, info=info)\n",
" loss = tn.norm(t3-1)\n",
"\n",
" if it % 200 == 0:\n",
" print('{:04}: loss={:g}'.format(it, loss))\n",
" loss.backward()\n",
" optimizer.step()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example 3: Fixed Grid\n",
"\n",
"Differentiable cross-approximation can be also used to find parameters $\\pmb{\\theta}$ such that $f(\\pmb{x}, \\pmb{\\theta})$ satisfies some property for all $\\pmb{x} \\in \\Omega$, where $\\Omega$ is a fixed (no to be optimized) grid of points.\n",
"\n",
"In this toy example, we will find $w_1, w_2, b$ such that the simple neural network $\\tanh(w_1 x_1 + w_2 x_2 + b)$ is as close to $0.5$ as possible for all $(x_1, x_2) \\in [-1, 1]^2$."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"N = 2\n",
"I = 64\n",
"domain = [torch.linspace(-1, 1, I) for n in range(N)]\n",
"\n",
"\n",
"# Super-simple 1-layer MLP\n",
"class Net(torch.nn.Module):\n",
" \n",
" def __init__(self):\n",
" super(Net, self).__init__()\n",
" self.linear = torch.nn.Linear(N, 1, bias=True)\n",
" \n",
" def forward(self, Xs):\n",
" Xs = self.linear(Xs)\n",
" return torch.tanh(Xs)\n",
"\n",
" \n",
"net = Net()\n",
"function = net.forward"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us visualize the network's response over the domain $\\Omega = [-1, 1]^2$:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Cross-approximation over a 2D domain containing 4096 grid points:\n",
"iter: 0 | eps: 5.079e-01 | total time: 0.0063 | largest rank: 1\n",
"iter: 1 | eps: 1.222e-05 | total time: 0.0261 | largest rank: 4\n",
"iter: 2 | eps: 1.377e-09 | total time: 0.0378 | largest rank: 7 <- converged: eps < 1e-06\n",
"Did 2304 function evaluations, which took 0.001031s (2.235e+06 evals/s)\n",
"\n"
]
},
{
"data": {
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",
"text/plain": [
"<Figure size 432x288 with 2 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"import matplotlib.pyplot as plt\n",
"\n",
"def visualize_net():\n",
" t = tn.cross(domain=domain, function=function, function_arg='matrix')\n",
" plt.figure()\n",
" plt.imshow(t.numpy())\n",
" plt.colorbar()\n",
" plt.show()\n",
" \n",
"visualize_net()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We now optimize the network:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0000: loss=33.3485\n",
"0200: loss=18.9223\n",
"0400: loss=7.43317\n",
"0600: loss=1.24353\n",
"0800: loss=0.0103478\n",
"1000: loss=0.00480166\n",
"1200: loss=0.00513987\n",
"1400: loss=0.00359209\n",
"1600: loss=0.00339219\n",
"1800: loss=0.00377836\n"
]
}
],
"source": [
"# Optimization\n",
"optimizer = torch.optim.Adam(net.parameters())\n",
"for it in range(2000):\n",
" optimizer.zero_grad()\n",
" \n",
" if it % 400 == 0: # Update cross indices\n",
" _, info = tn.cross(domain=domain, function=function, function_arg='matrix', verbose=False, return_info=True)\n",
"\n",
" # Forward pass\n",
" t2 = tn.cross_forward(domain=domain, function=function, info=info, function_arg='matrix')\n",
" loss = tn.norm(t2-0.5)\n",
"\n",
" if it % 200 == 0:\n",
" print('{:04}: loss={:g}'.format(it, loss))\n",
" loss.backward()\n",
" optimizer.step()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Finally, let's check that the resulting network satisfies the property we wanted:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Cross-approximation over a 2D domain containing 4096 grid points:\n",
"iter: 0 | eps: 1.167e-11 | total time: 0.0100 | largest rank: 1 <- converged: eps < 1e-06\n",
"Did 192 function evaluations, which took 0.0003726s (5.152e+05 evals/s)\n",
"\n"
]
},
{
"data": {
"image/png": 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",
"text/plain": [
"<Figure size 432x288 with 2 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"visualize_net()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.3"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
