explain-08.ipynb

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    "### Some terms you need to understand for understanding Large Language Models\n",
    "  - logits\n",
    "  - softmax\n",
    "  - cross entropy\n",
    "  - one hot encoding\n",
    "  - encoding/embedding\n",
    "  - tokens, tokenizer\n",
    "  - attention\n",
    "    - multi-head attention\n",
    "  - **positional encoding**\n",
    "    - rotary positional encoding (Used in LLama)\n",
    "  - *transformer*\n",
    "    - *encoder*\n",
    "    - *decoder*\n",
    "    - Residual Connection\n",
    "    - *Feed Forward Layer*\n",
    "    - *Layer Normalization*\n",
    "      - LLama suggests RMSNorm instead <https://arxiv.org/abs/1910.07467>\n",
    "  - optimizer\n",
    "  - Important players in the AI Ecosystem\n",
    "  - How to train/fine tune!"
   ]
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    "### What is positional encoding?\n",
    "  - Intuitively, we perturb the embeddings of tokens to encode *spatial positions/relationships* of the tokens to the embedding.\n",
    "    - The NN *learns* the spatial relationship between the tokens\n",
    "  - Experimentally, it seems to produce faster training (i.e. it converges faster)\n",
    "    - Unclear if more training without PE can approach similar loss values.\n",
    "  - How does it work?\n",
    "    - Magic! (err!! **Divine Providence**)"
   ]
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    "### Rotary Position Encoding <https://arxiv.org/abs/2104.09864> (for LLama)\n",
    "  - >Transformer-based language modeling usually leverages the position information of individual tokens through a self-\n",
    "attention mechanism. ... $q_{m}^{T}  k_n$ typically enables knowledge conveyance between\n",
    "tokens at different positions. In order to incorporate relative position information, we require the inner product of query\n",
    "$q_m$ and key $k_n$ to be formulated by a function $g$, which takes only the word embeddings $x_m$ , $x_n$, and their relative\n",
    "position $m − n$ as input variables. In other words, we hope that the inner product encodes position information only in\n",
    "the relative form:\n",
    "  - Intuitively, it is treating consecutive values in the embedding for a token as a point in 2-space and rotating it. The angle of rotation is proportional to $m - n$ the difference between the positions of the tokens\n",
    "  - Find a solution such that $\\displaystyle⟨f_q (x_m , m), f_k (x_n , n)⟩ = g(x_m , x_n, m − n)$\n",
    "    - i.e only depends on $x_m$ and $x_n$ and $m-n$\n",
    "    - $\\begin{aligned} f_q(x_m, m) &= (W_q @ x_m) \\displaystyle e^{i m \\theta}\\\\\n",
    "       f_k(x_n, n) &= (W_k @ x_n) \\displaystyle e^{i n \\theta}\\\\\n",
    "       g(x_m, x_n, m − n) &=  Re[(W_q x_m ) @ (W_k x_n)^{∗} \\displaystyle e^{i(m−n)\\theta}]\\\\\n",
    "       \\end{aligned}$\n",
    "\n",
    "    - $(W_k @ x_n)^{∗}$ is complex conjugate of $(W_k @ x_n)$\n",
    "      - $conj(x + iy) = x - iy$\n",
    "      \n",
    "  - R can be of this form\n",
    "    - $f_{(q,k)}(x_m, m) = \\begin{pmatrix}\n",
    "                               cos(m \\theta) & -sin (m \\theta)\\\\\n",
    "                               sin(m \\theta) & cos (m \\theta)\\\\\n",
    "                             \\end{pmatrix} \n",
    "                             \\begin{pmatrix}\n",
    "                               \\displaystyle W^{11}_{(q,k)} & W^{12}_{(q,k)}\\\\\n",
    "                               \\displaystyle W^{21}_{(q,k)} & W^{22}_{(q,k)}\\\\\n",
    "                             \\end{pmatrix}\n",
    "                             \\begin{pmatrix}\n",
    "                               x^{(1)}_m\\\\\n",
    "                               x^{(2)}_m\\\\\n",
    "                             \\end{pmatrix}\n",
    "                             $ (Eqn 13 in <https://arxiv.org/abs/2104.09864>)"
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   "source": [
    "import torch\n",
    "from torch.nn import functional as F\n",
    "from torch import nn\n",
    "from tinytorch.MyStuff import *\n",
    "from tinytorch.tensorhelpers import *\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "from matplotlib.patches import Ellipse\n",
    "from matplotlib.text import OffsetFrom\n",
    "from subroutines import *"
   ]
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   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "def rot2d(theta, device='cpu'):\n",
    "  return torch.tensor([[math.cos(theta), -math.sin(theta)],\n",
    "                       [math.sin(theta), math.cos(theta)]], dtype=float, device=device)\n",
    "\n",
    "PI = math.pi\n",
    "PI2 = 2*PI\n",
    "def toRAD(degree):\n",
    "  return PI * degree/180.0"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "pt = torch.tensor([1.0, 0], dtype=float)\n",
    "m2 = toRAD(15)\n",
    "m3 = rot2d(m2)\n",
    "print(hprt(m3))\n",
    "Rtn = [pt]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3d07514a-5191-497b-a706-950139aca7ba",
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   "source": [
    "xs = [ t.tolist()[0] for t in Rtn ]\n",
    "ys = [ t.tolist()[1] for t in Rtn ]\n",
    "fig, ax = plt.subplots()\n",
    "plt.axis([-1.4, 1.4, -1.4, 1.4])\n",
    "ax.scatter(xs, ys, marker='s', label=f'[1,0] rotated {(len(Rtn)-1)*15.0} degrees')\n",
    "ax.axhline(y=0, color='k')\n",
    "ax.axvline(x=0, color='k')\n",
    "#fig.set_size_inches(4, 4)\n",
    "plt.axis('equal')\n",
    "plt.legend(loc='upper right')\n",
    "plt.show()\n",
    "Rtn.append(m3 @ Rtn[-1])  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f74685ab-f752-4572-bb65-af6471c8ee12",
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   "outputs": [],
   "source": [
    "print(hprt(rot2d(toRAD(0))))"
   ]
  },
  {
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"
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wPbLnz5/v2bMHALBo0aLXr1+fPn0a9vHz85s1a1ZaWpqxsfGKFSsEAoGJiYmpqamNjQ3xUEqWeOjZrBDq6upA+y8ocA4dOjR79uxVq1YdO3Zsw4YNy5cvv3z58u+//453ePPmDZfLJXyi3cAToLa2lhygFOl0UCYPQhJ0lh3y5s2biIgIBweHNWvWzJ07NyIiwtnZedeuXdHR0cRu7bXAJEFnHZTDg5AEJYKjWi8EeCa0N8kgov1mq6qqtm3bFhcXt3379u+//17qig+PD08G2SGdDnfv3n337t3QoUPh7owZM2JiYlRUVGJjY/EF/SGWlpZwcPPnn3+eP3++sbGxo6Pj48ePp0yZgvcxMTGZP39+RkbGwIEDx4wZAxsfPHhgbW39+eefw92bN2/iJ9WTJ08MDAzs7OxGjRo1atSoioqKsWPHwhcv09LS4ALLFRUVsHN7obPsENHWsmw2+48//nB0dNy4cSP5k+2BzjqsWbPm9OnTenp6s2bNCgwMFHFlTcxGDQ0N+POAo3yJh2q9EGAJbm+SQaZPn/7ll18CALhcbl5eHrwrnDJlChzU27t378yZM52cnH777bfKysoDBw6QPi4h8He+ubmZHKAU6XR4+/YtEPapmpoa0pqXqqqqcBTSzc0tLy9v48aN1dXVtra258+fJ/ZRVVVNS0sj3l9fu3YNtwcSCAQxMTH4+uYlJSX4K/UYhoWEhOALtY8ZM2b9+vVMJrODl1f0lB3i6ekJv2RWVlaPHj2GDBkCAPj+++/hcxEtLS3c9Zf0wXZBZx2srKyKi4v9/f27deu2fv361qPwOKKf1ipf4qFaLwSoaXsHCiFqav9/BHLz5k1dXV3SqoGWlpbwiRMAYMiQISL83kQDTwDSlQjlSKeD8nkQ4tBZdgiefunp6bhEOB20wMShsw7K5EGIQ4ngqNYLASZZey8oSAj1m127du2cOXMAAM3NzfHx8Q4ODsSPSA48ATr4txeLdDoonwchDp1lJyHUWpYq6KyDMnkQ4lAiOKr1QuDz+QwGg+RLICES+s1u2LBh+/btIrw9RQNvPzt4TycWqXVQMg9CHJrLLtZalirorIMyeRDiUCM4XNoYQQS+lg3NzNrLN998c/LkyS1btqSnp9vb2yckJBw8ePDVq1fEPt7e3gkJCZg4vzcRzJw5EwBw8+ZNcoBSOqIDplwehBCay15aWgrtSbds2XLp0qVVq1ZdvXr10KFD0BETpyMWmBA666BkHoQQSgRH8+uFgGEY/t/2kpmZ+fz58969e3fp0sXa2vr169fTp08ndjh06NCcOXPMzc1ra2uDgoK2bNlCjEoI/G4CgYAcoJSO6AAAYDKZcIk+EgYGBqS7HHifPmDAAGIjEZIVHCECAACt17zt378/qYUSaC577969Hz58+OrVq4ULFzIYjG+//baurm7GjBnkfh2GzjrAXOrRo0ePHj3IsfajTImHxnCoR4Tf7K+//rpp06YRI0YwmUw9PT0tLS1iFCFTOmIFpyiItpbtuAUmQgpoknjoul6u+Pj4+Pj4kFsRcmHy5MmTJ0/evHkzOfDJwGQyPT09PT09yQGELKFJ4qHregQCgVB+UK1HIBAI5QfVeoTSQq0VHAIhIfRMPFTrEcoJ5VZwCIVGcT0IqQLVeoQSIgsrOIRCo7gehFSB5uEIgfguNT15//49kP33lPXxZYcsrOAAkv0DiqiD4noQAooER7VeCBwOBwBAiX+YjICLo+LLqMkI+usgFBlZwQEk+wcUUQfRq1pSAs0TD9V6IXTv3p3D4aiqqpIDtOHzzz8vKiqi0GlIKB3RQfk8CIEiyC4f6KwDX+k8CAFFgqNaTz082XsQ0hxl9SBUXORgTEgTlNKDkBLQs1nqEetB2NDQ0NTURPiEUkFnD0I+n19ZWdneJVYUHQmNCd+/f0+reSPSQU8PQjokHrqup5i2PAjhKmA1NTWhoaFffPHFixcv4uLiAgICdHR0yIdQcFp7EMLlt4R6EP7555/Hjh2DHoSGhoZTpkxZsWIFyYPQxMQkMjKyvVZwcDcmJga3ggsKCqqrq9PR0blw4cK6deugkcCnQGBg4I8//ghaGRMSB8oAAKtXr54wYYKrqyuxUeFYs2aNjY3NiRMnJk2aNG/ePBcXF3KPDxCzUagHoZIlHqr1FDN9+nToDUTyIITRuLi41NRUDw+PiRMnHjx4MCYmZunSpcSPKwHSeRBOmDAhOjo6MTHR1tb27Nmzy5Ytw/sAAMRaweFWVq2t4OCtQ1lZWWRkJLyS7du3r729fU1NTeuVCxULZ2dnqLZQ5s+fDyudp6cnzEmSMSGxc3R0NG4Xo9BAD8LLly+npaWtX7/+xo0bRNscIqKf1ipf4qFaTzGiPQidnJzgSGJVVVVJSQlueaNM4B6E+FBMU1MTj8ezsrK6fPkysSfRg3Dr1q3QhvDy5cvHjx/Haz0kLS0N95GAVnCbNm2Cu0QruBEjRrRlBcdkMnV0dFpaWlRVVYcMGdLY2FhRUSF0yWUFIjg4mNwkDNHGhACAkpISgUAwcOBAckAB8fPz8/LycnNzc3NzKy0tHTFixKlTp6A1lRQoU+Kh8XpZIdSDUEVFhclknj9/3sHB4erVq9CNQcmgpwehnp5eSEgIvFiLiooaM2aMqanp//8HnwxCjQn5fH5sbGxnDSxQDt08COmTeKjWU4xYD0JVVdVly5b9/vvvW7ZsycrK+u+TSgSdPQiLiorCw8NjY2Pl8GYNHRBrTHj+/HnSXZRCQ1sPws5PPAzRCjj4QPIbkxDRHoT379+vrq6G2y4uLtOnT//vk+0BPu3MzMwkByilIzpgtPQgLCgo8PDw4HA4bDZbin+XQshOQrQxYV5enrOz85EjR44cOTJ+/PhFixZdunSJfIhW0FkHenoQ0iHx0Hi9ECZNmlRVVSWdDbxoD8L169e7uLjAJ2Zqamr6+vp4qF1YWVnxeDxDQ0NygFI6ogOgnwdhXl7e1atXDxw4gGHYmTNnlixZ0vpQolEI2UmINiYcPHgw/nTx9u3bU6dOhVepoqGzDjT0IKRL4pGLP0JaU2NJSElJ8fX1zcrKOnfuHPwlIPeQDEq8hsUiOx3kT2lpKfFU0dfXJ/eQACWW/fDhw5aWlvb29pcvXybHWqHEOlAOfRIPXdcLAZPK1FgSbGxsxo8fX1hYOH78eGdnZ6lH7uB366DXsFhkp4P86d27N5zf2RGUWPZ2eRMqsQ6UQ5/EQ7Ve3nz22Wf4PFwEAoGQD2geDgKBQCg/qNYjlBZ6WsEhlB56Jh6q9QjlhLZWcIhOAXkQolqPUELobAWH6BSQByF6NotQQmRkBYdQXBTag5ASZFLr6+rqgoODdXR0nJ2dyTEEQsbIzgoOobiIXtWSEmieeDKp9XBN9ps3bypore+gh68coMRrWCwdOb5SehDSX3b5QGcdlNKDkBLBZVLrAQC5ubmTJ08mtyoInI6ZGvNk70FIidewWKTWQVk9CGkuO0QOdoN01kEpPQgpEZziZ7MVFRXh4eElJSXp6elwCUNFBA63tdfUGEesB2HHgc44FJ69QpFOByX2IKSz7BAJ7QY7CJ11UEoPQkoEl+i6Hi49SGxhMBgkjyEAQHR0dEpKio+Pz5EjRxgMBr4W6CeFaA9CIq9fv1bK9euRB2EnIondYENDg5qaWnvXFFMUkAdhW4iv9bW1tdOnT6+oqCgrK2MwGMOGDWtpaYFLfQ4YMMDBwWHt2rWqqqpVVVVr164tLCzU1NQ0MTERan/zKSDagxDn3LlzqampFy5cILUrAciDsBMRbTf4KdgdIw/CtiBfm7dGV1c3KyvLwcEBADBq1KiHDx/m5uZWVFQ8efLks88+W7duHXymcfXq1dGjR8N/Q0ZGxidb60V7EEKePn2akZFBbpUxPB4vPz+fy+WSA1SDexDiLU1NTfX19VZWVqWlpf/1+9iDEBoQJiYmhoWF4ecPTlpaGp5R0AoO3yVawTU2Noq1ggMA4FZwsJtMaW5urq+vJ7fKDNF2g9DueOLEic7Oznl5eXCgWT7ILf38/Pzgg9bQ0ND8/Px///2XdMHeLhQ38VojvtZD0tPTAQDEx63dunWD9+ahoaF8Pp/D4cBbGy6Xm5CQYG1tHRISgnf+BBHqQQgAaG5uTkxMtLe3/6+rjLl169ahQ4ciIyNramqSkpKOHTsWHx9P7kQdyIOQSFpaWqfcvQm1G3RycoJPDuVpdyzn9EMehG0hfgwHAFBXV3f//n0AAOlKAVro9evXT01NzcHBYdWqVTExMS9evBg+fHhsbKxyWBW3l7lz544ZM2bjxo1///330qVLYWNMTAy8oQMABAUFubi4yDTdifj4+IwfP97Ly4vY+Pjx4w0bNhw+fJjYSCEBAQGenp7u7u62trbFxcWDBw/u2bMnACA1NXXbtm3Tpk3T1ta+cuUKyYNw6NChampqV65caa8H4aVLl1pbwV29epUOVnAYhrUuIjKitrZ2zJgxZ8+e1dfXz87OhlNQoN0gNMzB7Y6DgoLkY3cs//SDHoRNTU1GRkYpKSm4B+Hp06ehB6GPj4+fn19cXByPx9u4ceOePXu2b9+elZX19OlTNTU1og2hQieeEGAuigYONqmqqtbV1eGN9+/fV1dX79atW1paGmwRCATv3r2DG8SeHeSff/5ZuXLl3Llz582b9+LFi507dzo6Oq5atUpqow+xSGd+BhHtQZiRkXHv3j0Mwy5duuTo6PjRJ9uDhJ5kKSkpV65c+fnnnxsbG/FGgUDg6+ubmJgYGhpK6CuEjuiAIQ9CDMMwLD4+/vjx4+RWkUgtu2i7QSK5ubnjxo27e/cuqV1CJNShU9IPeRC2hUTX9WlpaQCAXr16JSUlYRhWX19///79f/75Z/ny5V5eXvj1O4PBgI96GAxGt27diEeQmtOnTxcVFf3xxx+qqqp9+/Y1Nzf/5ZdfVq9ePXPmTD09vf3795P65+bmpqSkkBqJqKqqurm5yW4SgggPQgzDjh49OmHChGvXrj148KCgoODo0aMeHh4fH4BKbt269euvv37zzTc7duzw9vbW0NDAMMzX19fBwcHExOTQoUPNzc2in1B1BOXzIKQ5ou0GAQAPHjzo3bu3oaHhkCFDBg0atG3btoSEBMIBKKZT0k8pPQgpQaJaDwfrx4wZo6am1tLS8urVqzNnzhgbG69evVqmAzV8Pv/cuXOZmZmqqqp8Pv/t27cMBmPDhg3R0dG6urpCpy4ZGhris/2EoqqqSnl6kcBrloqKCv5OHQCAwWBERkbC7dOnT/P5fJkW+rdv3/br1w8AoK+vv3nz5h07duzatevIkSNLliyB32rMmDF5eXlUvdpHc8rKyiZPnlxTU7Np0yYAgL6+vkzF7yzU1dXx4WAmkwlfccChyu5YElD6QeiTeOJr/bt37x48eAAAcHJyguUVjqUEBQX9+uuvV69eJfVPTk6urKwEADCZzFmzZnXkClpNTe3WrVtweCsnJ4fL5U6YMEFbW3vJkiVLliwh9wYAANCzZ084NNwRJkllaiw5ly5dunLlSmlp6bFjx9atW0cOS4AkXsO5ubn4LCB9fX0vL6/Ro0eHh4fjlx5Dhw6NiYkRcbLJWgd5QokVnCSy+/v7p6am4rvV1dUNDQ3ExzOWlpZ+fn74bmtkJ/uOHTuys7Ozs7Nzc3OLiopCQ0PJPSRDEh1Q+kHklnjiIQ7oCOXKlSsAAAaDQRy6mjt3LgDgyy+/JHTE6urqHB0do6Ki4G5BQcHChQtzc3OJfaTG19cXALBz505yoBXvRSJ0NI0E/U2NJfEafvnyJf63aGlp2blz58OHD7du3crlcmHjw4cPr1279t8HWkF/HeSMJLKTkGK8XqayNzU1PXz48OnTp0Kfi0iIJDqg9KMQSQQXi/jrejiAM2LECOLQFXyfGB/AEQgEKioqHh4eX331FT6b0MLC4pdfflm0aFFWVhbpXlJCBAKBh4fH4MGD16xZAyekf/PNNzD0+++/W1lZ4bs4ycnJ+KoXQlFVVTo6poEAABo+SURBVD137pzo74PR3tQYfjfRXsP9+vULDw+3t7cXCAS7d+92dXXt169fv3794MQDJpN5584d+NpEW9BfBzkjiewdR6ayU2J3LIkOKP0oRBLBxSK+1sMHs8TZls3NzXAmKXz6kZ6eHhkZ+fPPPwcFBcEZTjgDBgxgMplhYWGt31SOiYkxNzc3MzO7efNmTU2Nra2tmppaWlpac3OznZ0dHLe5cePG8ePHV65c+eLFi6SkJAAAfNZXUFAQFhbWel06AMDUqVOnTp1Kbv1U6d69+7Nnzy5cuADPNNiyefPmPXv2rFu3rqGhQUtLi/wZBIIiUPrRClG1fsaMGWVlZcXFxUwmMyIi4vr16wkJCfr6+urq6jt27Ni2bVt8fPzhw4ejoqLOnz+fnZ3dq1ev+Pj4sLAwfLmi/v37W1hYZGdnk2p9VFTUwIED7e3tf/jhhzlz5rBYLEdHRxsbm8WLFx84cODt27fwFflhw4ZNnDhRU1PT3d09IiLiwIEDP/74o7m5eXV1dWRkpIaGBvGYiNa4uro6OTl5enrCMw2io6Ozfv16Nzc3Odj0ID5lUPrRClG1XsT7Pps3b3Zzc3v06JGGhsaPP/742WefFRcXczicPn36uLu74y9qv337NjIykjgXBaKrq6unp8disVasWKGjo1NQUPDmzRt3d3cGg0F8n6179+7p6elw2SAAwJw5c9hstrq6uugRGAQOg8EIDg4OCQm5d+/ezJkzjY2Ny8vL09PTm5qawsLC0I8lQqag9KMVomq9aPT09IgDO6NGjQIADBw4kDhLGsOwHTt2/PTTT3gLxNra+vLly99++y0s4tevX4czggEAqamppP7E5Zmomrb/6aCqqurs7MzhcO7fvx8XF2dqajpr1iy4WixCDlhZWXXWO/F0AKUffZB0PRyxdOvWzd/ff9u2bcTGs2fPfv311/gqj/Hx8fiSFMT10VJTU+Ficrm5uUwms2/fvvh6/whK0NLSsra2XrVq1ZQpU9CZJk8MDQ2FvlD2SYHSjw5QVusBAK6urvPmzVu/fv21a9dycnL2799fXV2Nr1nI5XJdXV3hDE4AwIMHD+BvAJ/Pr62thVNxu3Tp0qNHj2PHjuHvmiIQCASi40g/hiOUBQsWLFiw4PXr1w0NDZs3byauV961a9fnz5/jy6gmJyfD91fV1NTy8/Nh48CBAy9dutSlS5fWRigIBAKBkBqKaz2krfXzzp49i1vQERcqIG7T4blrBz185cB7KryGxSLr4yscSHYI0kHOUCK4XC+fv/32W4V4uMqRytRYnlDiNSwW+usgZ5DsEKSDnKFEcLnWeqGLwNEQ6UyN5QklXsNiob8OcgbJDkE6yBlKBJdrrUcgEAhEp4BqvTwoKChwdHQkua0iJMfb2/v48ePkVoRkoPSTmsbGRhcXF7hOjKKDar08uH379sWLF3GrVUR7OXv27Pnz58mtCMlA6Sc15eXlFy5caL1yuyIik3k4CBLLly+fMWNGr169yAGEZDx69Ag6RCOkAKWf1JiampaXl1NicdXpoFovJ9CZ1hGI62QgpACln9R03PuIJqAxHAQCgVB+UK1HIBAI5QfVegQCgVB+UK0XwqRJk6ZNm0ahqTHlk96srKxsbGw66jUsDsp1kBqazLlUUNlR+kkNTeZcUiM4hmgF5abGgYGBAIDY2FhyQFoo8RoWC+U6SE2/fv2++uorcqvcUVDZUfpJzbNnz1RUVDZt2kQOyBdKBEfzcISAUW1qTPmkN/jdOug1LBbKdZAamsy5VFDZUfpJDU3mXFIiOKr1coLCM+0TBM257CAo/aQGzblEIBAIhMKAaj0CgUAoP6jWIxAIhPKDar08oHzS26cGTeZcKigo/aSGJnMuKQHVenmAFhrsIGidy46A0k9q0DqXiPZB+aS3Tw2azLlUUFD6SQ1N5lxSAqr1Quigh69QqD3TKPEaFousjy85NJlzqbiyo/STGjrMuaREcDSGIwT6mxpT4jUsFvrrIGeQ7BCkg5yhRHBU64VAf1NjSryGxUJ/HeQMkh2CdJAzlAiOaj0CgUAoP6jWIxAIhPKDaj1CGSgtLU1PT+/gwysEor0oUOKhWo9QeE6cOBEaGtq3b18HB4fOXRYR0em0tLSsWbMmJSWFHJABipV4aM4lQrH5+++/4+Pjo6OjAQAsFuv27dvjxo0jd0J8MtTV1YWEhOjq6k6ZMoUcoxSFSzxU6xEKDIvFcnd3T01Nhbuampo5OTk0P+UQMkVPT6+6uprJZJIDlKKIiYfGcBAKzJkzZ4YNGzZixAi4W1FRgaZjI2Rd6IFiJh66rpeSsrKya9eulZWVTZgwQU9PLzk5uaWlZf78+R2cA6uIBAYGVlZWGhsbNzU1LVq0CE6L5nA4AQEBOjo6fD5fXV19+fLlDAYDAPD48eOsrCxtbW02m33v3r0dO3YQ30tMS0vLzc0dO3bs8OHDw8LCuFxucXHxli1b1NXVk5OTq6ury8vLt27dCk9mgUDwxx9/bNq0CX6Wz+cXFhYOHDgQPxoCJysr6/nz58+ePVu5cmV+fn5JSUlTU5Obm5uamgJXABaLFRkZaWhoWFdXV15ePnLkyG+//Xb37t2FhYVjx4718PAoLy/fuXPn27dvt27d+vbt29LS0pcvX06aNGny5MmkQ30SiYchWtGrVy9NTc36+npy4AMsFsvf3x/DMA6H06tXr3379nG5XAMDg4CAAHJX2fDVV1917do1OjqaHKAUsTpgGObq6rp27Vq47e/vv2jRIgzDamtrv/rqq9zcXNgeEhIC2xsbG21sbD58FHNycnrx4gW+++bNmz/++CMnJ2fYsGG//fZbXV0dhmHe3t5OTk6nTp3i8XgYhi1btuzgwYOwf2JiIgDAx8cnJCQkJCTE19cXAPD06VP8gCJobGwUCATkVnHQR/Z2kZmZGRcXh2FYXFzcqFGjIiIibt26paamVlRURO4qGTTR4bvvvquoqIDbZ8+eDQoKwjCspaXF3t5+5cqVsL2+vl5XV3fHjh0FBQUYhhUVFWlra7979+7DMTDsk0k8NIYjBHV19YaGBhHzqAIDA1evXg0A4PP5FRUVs2fPZjKZJ0+eXLJkCalnSkrKgQMHSkpKSO0iqK+vJze1onv37vDXhRygFLE6ZGRkBAYGent7w10ul2tmZgYA2LZtm4WFxZAhQ2D70qVLb926dfny5Tdv3uTn5xcWFsJ2Z2fnrl27wm0AQHR09OzZs589e1ZVVbVy5cpu3boBAJqamrKzs11dXdXV1QEAGIax2WzY/+HDh71797a1tbWwsLCwsCguLjY0NIRvGOLU1dUdO3YsODiY2AgA2L59e3V1Nb5bWVkp4p+JQxPZ20tKSgo0p+bxeK9fv543b96gQYMiIiLMzc1JPYODg318fMT6mtJEh8ePH9+5cwfDMACAvb19nz59AAAqKiqDBg3C+2hpaZmYmNTV1VlYWAAABg4cyOfzSUt+fiKJp8B3cLIDLqnY3NxMDnzA09MT3vxmZWX16NEDFrXvv/+e3A+Ar7/+etmyZcuWLSMHWsFms0NDQzMzM7t27RoQEEAOfwwcHIRZKDvE6pCQkGBkZKSvrw93f/nlF7gRGxvr5ub2Xz8ALC0tY2Nj582bN23aNEtLSwsLi6lTp65fv97Q0BDvs3jx4i5duqSnp8+ZMweebwCABw8eODg44P/SmzdvLly4EG4/efJk3Lhxo0aNwnddXV1Jr9TDRdNu3rzp7OyMN75///7Nmzdw7OjGjRtpaWmff/75v//+O3To0J07d+LdWkMT2XGcnZ3fvn1Lbv3A/PnzXVxcAOHvkpWVNWnSJBUVFV1d3blz5xI7Q0aOHLlv375ff/2VHPgYmujw888/z58/39DQcMqUKStWrBAx8QYWegAAg8Ho0qULj8cjRj+RxEO1XghQUxFJho9ypqenT5o06aPYx9TW1mppaRkbG5MDrdDS0nJxcWEymZmZmeRYK+B36+DfXixidXj79i3xwhyHxWKpqHx0y6iqqgovZ86fP79y5cqkpKSYmJjAwMCcnBz8POzSpQsAIC0tDc97gUBw48aNzZs3w928vLzXr19bW1vD3ZKSkgkTJsDtly9fZmZmBgYGwl0iubm5pPHZiIgI/IfZ29v7woULPXr0mD17ds+ePa2trW1sbIididBEdpzWl41CIabr0qVLPw5+RG5uruh8htBEBzc3twkTJkRHRycmJtra2p49e7atiyrRX/UTSTw0hiMEsRcUOKmpqfhfHY7iQQQCQUZGRlRUVGpqqiQnDwBARUVFQ0OD3NoGTU1NAAChdZZCxOpgZWX18uVLYoc3b94AAEaOHElyQSotLYWdQ0NDJ0yYsGfPnvv377u6uoaHhxO7VVZWFhYWTpw4Ee4+ePDg/fv3o0ePhrsXL160s7PT0dGBPkGampr4Xe3p06cdHByI99EVFRXh4eElJSXp6en4ASEJCQm2trZw28zMDN6ba2tr9+vX7/nz58SeJGgiu3Q0NDTcvXtXaLo2NDRER0ffunVL7LULhCY67N+/f9CgQV5eXomJiWFhYUFBQeQeEvMpJB6q9ULQ1tYGADQ2NpIDAAAAamtrzc3Nr1+/np+fn52dDS9L6+rq8vPzYYfGxsYlS5ZoampaWlp6e3tLcvK0l4aGBgAAfr8pI0TrAABwcnIyMTEhOkYdPXoUALBnz57Y2Fi4Ji0A4NGjR1VVVT/99BOHw/Hz88M7Dxo0SE9PD98FAKSlpZmbm+OLrScnJ48bNw5edgEAMjIy5syZw+Vy79+/DwAYN24c1OHp06dRUVH+/v4fDgOio6N9fX3t7OxCQ0MZDAbxvqqgoMDS0hK/7Thx4gR8xvDkyZOqqio7Ozu8Z2toInu7yMjIMDExaWxsDAkJEQgEcCz79u3bcJgbAPD06dMVK1aMHj2axWKFhYWRypNQaKLDiRMnqqqq4DYxl3g8HnEQnMfj4YM2GIY1Nzfj/3acTyHx0BiOEHR1dQEAdXV15AAAAAAul6umplZeXh4XF/fXX3+Fh4c3Nzfn5eX9+OOPsIO3t7elpeWXX34JABAIBPjJc+XKlXfv3uHHwZk2bZokgzxEYBmFJ4PsEK0DAIDJZCYlJbm7u9fU1JibmxcVFTk5OQEAJk6cePTo0eXLl8+bN4/FYqWnp6ekpBgYGLBYLG1t7WPHjg0bNqympiYvL+/w4cPEA+bm5k6fPh3fzc7Ohg8VIVOnTn3z5s2pU6d++OEHAMCqVavc3d0TEhL++uuvqKgo/LFBVVXV2rVrCwsLNTU1TUxMSL+1gYGBGzZsILYAABoaGjw8PJKTk0V7etBE9nbBZrM///zzS5cu8fn8zZs3//7773369GlsbMSHO5YtW3bgwIGePXuOHDnSyMhIklSkiQ5GRkbBwcFDhw5VU1O7cuXKvn37AAA7d+6MiYlpaWnx9fV1cnLy9fV9+fLlmTNnMAybOHHi4cOHMQzz9vauqqoiDmd9Eon335QcxAdgwYqPjycHPsDj8Z49ewbnTnG5XHziF8TExOTRo0cYhpWWlpqbm+Ptubm5t4XBZrPxPkFBQfh0MRHo6uoymUxyK9WI1QGHxWK9evWK3IphJSUlNTU1+K5AIOByufCnsbq6mtDx/zQ0NDQ2NuK71dXVpAlqZWVlxF0+n19cXEzqc+7cuXnz5sHtlStXhoaG4qHGxsbW8tbX169du7a4uBjDMNKfkgTdZJeQurq6169fw+2amhriLMaSkhJ9fX0oYEhIyKpVq/CQCGiiQ0NDA4ZhxcXFL168kGIiI5FPIfHQGI4Q4OQQodfgEHV1dVNTU/hyEJPJJL0/xeFw4Pjd5cuXra2ts7Kynjx5AgAYMmTIGGG09+e6paXl3bt3HZyAJQlidcDR19fv27cvuRWA/v37E78ng8FgMplqamqDBw8mzsDB6dq1K/GhhaGhIRQZh3TVqaqqOmDAAFIfXH8ul5uQkGBtbR0SEgJDxIdjEDab7evru3v3bmNj43v37l2/fp0YJUJD2SWkW7ducD4iAMDAwEBLSwsPcTic/v37QwH/+ecfa2vrixcv4lGh0EcHOH49YMAAExMTUg60l08h8VCtFwI8MUTMZhPNjh074AJ479+/b2xsTEhIaD2RuTWNjY3+/v4RERF37949fPhwcXExuccHWCwWhmGS3Gt3kA7q0Fk4ODiUlJTAeT7Dhw+PjY3FtUpKSpo2bRqx88yZM/fu3WtoaMhkMkeNGmVpaUmMElFK2YcOHWphYREZGRkQENCnT5+CggJobSoCpdSBEmieeGi8XggwycrLy8kByVi3bl19fX3Xrl1VVVU5HI6ET8+ZTKanp6enpyc50IqamhoAAHywI1M6qENnoa+vf+nSJTabPWvWLHd39/r6evhQq6CgYPDgwaRrsRs3bhB3RaCssoeEhNTW1uKD42Jt3JVVh45D88RD1/VCMDExAQC8fPmSHJAYbW1t+G6FlpYWaaZ5x4GX/KampuQA1XRch86CwWDAmsVgMPDZC4GBgcuXL/+oX3tQYtlhoQcf3gASjRLr0HHonHgUlyHlAM5Le/XqFTlAD+AaAx3/nRcLzXVoFxiGNTc3E9dZay9IdgjSoV3QJ/FQrReCjo5Or169aHtBAf/2Hf+dFwvNdSAi1gqOwWAQ50FLAZIdgnQgokCJh2q9cEaMGPHq1Sv4CgPdKCoqUlVVHT58ODkgA+isA46EVnAdnKqBZIfQXAcaehDSJPFQrReOlZWVQCAgrYdHEwoLC4cOHUqcOSc76KwDBFrB/fzzz6amptAKjtyDIpDsEJrrAD0I5VDrFS7xUK0XztixYwEA8JVoWlFdXV1bWys3tzPa6gCBVnB79+6Fu9AK7uMu1IBkh9BfB+hBCN+hlR2KmHio1gtn/PjxKioq7Uoy+XDnzh0AwNdff00OyAba6gCRmxUckh2iEDookwchhYKj+fXC0dXVtbKyunbtGjkgATK1J0xISAAAyGI9NaFIosOn4EFIN9llmmMioJUOn4IHIZWC/7dcAuJjdu3aBQAoKSkhB0Qia3tCMzOz0aNHk1tliWgdFNSDsL3QSnZZ55gIaKWDInoQthcKBUdjOG2yaNEi8PEy35IgU3vCkpKSZ8+ezZ8/nxyQJSJ0oL8HYVu8f/8ea3viBAm6yS7THBNhgUk3HejsQdgWnZl4H1V+xMeMHDly7ty55FaRwNWxMQxLSkrq0aPHx8GP4HK5xsbGpPXzRHPq1CkAwJMnT8gBGdOWDlu3bu3Vqxe5FcP69+/v4+NDbLG1tV2xYgX2YfFCCwuLH3/88dmzZ8Q+8DRYs2aNm5sb3mhnZ+ft7Y3vmpqa4g7Lbm5uCxYswEOTJk3asmULvgt59+7d0aNH4RUfES8vr8rKSny3oqIC/8O1hm6yyyLH6urqTp06tXTpUqL4JOimQ0BAgKqqqpGR0dKlS1NSUvB2Ly8v4qKSX3zxxalTp/Dd7t27Jycn47vYJ5N46LpeFEuXLo2Li4NeSxIiC3tCnOjo6K+//lpGI4MiaEsH6TwIMzIyFixYkJmZOXz4cPwaHxCs4HAzNoFAcOPGDVzG1lZw+KRjaAW3atUquIuD234SG0m2n3v37k1JSXFwcNi9ezexGw7dZJdFjkELzGnTpmFtX3XSTQc3N7e8vLyNGzdWV1fb2toSPXNIiHbv+1QSj1z8EQTevn2rqal59OhRckACxo0bd/z4cbidkJCAt7e0tKSnp1+5cuXChQsSLhcOKS4uVlFRiYiIIAdkT1s6nD59Wk1NDQ5oQuCafNbW1u7u7v/1w7Dhw4fv2LHjxYsXFy9exBt/+umn3bt3E3phFRUVAIDy8nK4m5OTo6GhAS+7MAzbunXrd999h2FYamoqhmFz5sw5efIkHnJycoLbJNzc3IgriWMYdv78+bi4OLg9ffr0qqoqDMPYbDaTySReHkJoKDsO5Tkmwj6Bhjrs27cP346MjJw8eTLcbn1dD71KIK2v67FPI/HQdb0odHV1XV1dz507Rw60gUztCY8fP96nT5/vvvuOHJA9belAWw9CQJ3tJ91kl2mOiYBuOgBaehACGicemnMphvXr1586dSojI0MSH06p7QnFwuFwAgMDt2/fDpfPlD9CdWDS0oMQABAdHZ2SkuLj43PkyBGGONtPuCHU9pOGskuXY+/fvw8PD29d41RUVBwcHPBBobagoQ6Afh6EgOaJ99FVPkIY69evnzhxIrm1DaSzJxTL8ePH+/XrR7RJkz8idKCVB2FlZWWfPn04HA6GYa0HMby8vPBbdRwOhzNjxozs7GxSOz1llyLH+Hz+nTt3yO6Xt2/fvXuXqF5bYzj01IFuHoQ0TzxU68VTW1urr6+flpZGDrQffX196PZ57NgxV1fXu3fvFhUVkTu1gsvl9u/f/6+//iIH5AuFOsgUqmw/FVR26XIMIrTWK6gO8ofmiYfG68XTvXv3ffv2bdmyRSAQkGPtRDp7wr179w4cOHDx4sXkgHyhUAeZQpXtp4LKLl2OibDAVFAd5A/dE49c/BFtMHv27GPHjpFb2w+bzebz+RiG1dfXt7S0kMOtKCgoMDAwEDpC0ilQpYPsYLFY8+bNi46O/v333+3s7P744w98koOTkxPpvpu00sjjx49hu0LL3t4cE4FC6yBnaJ54qNZLCovFGjRoEPGdfvkwefJkyu/mOkJn6dAuBAIBfA9eIBDAV94xDMvPz9+/f/9H/doGyQ5BOrQLOieeKlxuAiGWrl27Qm94/JUKOZCdnc3lctetW0cOdB6dokN7YTAYGhoacAOfOefn5+fu7i7JOuBIdgjSob3QOfEYWKtpWAiE8oFhmKen55EjR8gBBEKW0CfxUK1HfCpgGNZBNzgEQgpoknio1iMQCITyg+ZcIhAIhPKDaj0CgUAoP6jWIxAIhPKDaj0CgUAoP6jWIxAIhPKDaj0CgUAoP6jWIxAIhPKDaj0CgUAoP6jWIxAIhPLzP+9GeWWUffMdAAAAAElFTkSuQmCC"
    }
   },
   "cell_type": "markdown",
   "id": "6b55bbc6-cdc9-4b09-b97c-fc022f461fad",
   "metadata": {},
   "source": [
    "### Rotation matrix, continued\n",
    "  - >To generalize our results in 2D to any $x_i \\in R^d$ where $d$ is even,\n",
    "    >we divide the d-dimension space into d/2 sub-spaces and combine them in the merit of the linearity of the inner product, turning $f_{q,k}$ into:\n",
    "  - $\\displaystyle f_{(q, k)}(x_m, m) = \\displaystyle R^{d}_{\\theta,m} W_{(q,k)} x_m$ (eqn 14)\n",
    "  - This the required matrix $R$\n",
    "    $\\begin{pmatrix}\n",
    "      cos(m\\theta_1) & -sin(m\\theta_1) & 0 & 0 & ... & 0 & 0\\\\\n",
    "      sin(m\\theta_1) & con(m\\theta_1) & 0 & 0 & ... & 0 & 0\\\\\n",
    "       0 & 0 & cos(m\\theta_2) & -sin(m\\theta_2) & ... & 0 & 0\\\\\n",
    "       0 & 0 & sin(m\\theta_2) & cos(m\\theta_2) & ... & 0 & 0\\\\\n",
    "       ... & ... & ... & ... & ... & 0 & 0\\\\\n",
    "       0 & 0 & 0 & 0 & ... & cos(m\\theta_{d]2}) & -sin(m\\theta_{d/2}) \\\\\n",
    "       0 & 0 & 0 & 0 & ... & sin(m\\theta_{d]2}) & cos(m\\theta_{d/2}) \\\\      \n",
    "    \\end{pmatrix}$\n",
    "\n",
    "  - Therefore, the attention formula then becomes $q^T_m k_n = (R^d_{\\theta,m} W_q x_m)^T (R^d_{\\theta,n} W_k x_n) = x^T W_qR^d_{\\theta,n-m} W_k x_n$ (eqn 16)\n",
    "    - where $\\theta_i = 10000^{-2(i-1)/d}, i \\in [1, 2, ..., d/2]$\n",
    "    - $d$ is `nEmbed`\n",
    "    - $m$ and $n$ are the absolute positions of $x_m$ and $x_n$, respectively\n",
    "    - $R^d_{\\theta,m-n} = (R^d_{\\theta,m})^T @ R^d_{\\theta,n}$\n",
    "  - The above matrix is mostly zeros, so we can derive a much more efficient operation to generate the desired rotation.\n",
    "  - ![RotaryPE02.png](attachment:c916dc26-64f8-4472-82b4-5453529a9530.png)\n",
    "  - ![RotaryPE00.png](attachment:7d8755a6-ca0b-4f56-884c-7b22d4e7f43c.png)\n",
    "    - RPE is applied AFTER q and k have been multipled by the W matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "333eddb5-29d3-429e-ae2e-7269c33125bf",
   "metadata": {},
   "source": [
    "### So what happens if we do a dot product of TWO rotated points??"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "26e34222-bc29-4bad-b228-6910ada2b1ff",
   "metadata": {},
   "outputs": [],
   "source": [
    "pt = torch.tensor([1.0, 0], dtype=float)\n",
    "m2 = toRAD(15)\n",
    "m3 = rot2d(m2)\n",
    "print(hprt(m3))\n",
    "Rtn = [ pt ]\n",
    "Count = -2\n",
    "Faster = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dc9adade-ebf9-4dd1-b5c1-b8eee91c3162",
   "metadata": {},
   "outputs": [],
   "source": [
    "def setupDeg():\n",
    "  global Rtn, xs, ys, deg, outer, outerIdx, xso, yso, Faster\n",
    "  xs = [ t.tolist()[0] for t in Rtn ]\n",
    "  ys = [ t.tolist()[1] for t in Rtn ]\n",
    "  deg = 15.0 * (len(Rtn)-1)\n",
    "  outer = [ a*b for i,a in enumerate(Rtn) for j,b in enumerate(Rtn[:i+1]) ]\n",
    "  outerIdx = [ (i,j) for i,a in enumerate(Rtn) for j, b in enumerate(Rtn[:i+1]) ] \n",
    "  xso = [ t.tolist()[0] for t in outer ]\n",
    "  yso = [ t.tolist()[1] for t in outer ]\n",
    "  print(f'setup {len(Rtn)=} points, {len(outer)=} dot products')\n",
    "def drawDeg():\n",
    "  global fig, ax, currDot, currPoints, cxs, cys, Count, Faster\n",
    "  global Rtn, xs, ys, deg, outer, outerIdx, xso, yso\n",
    "  currDot = 'None'\n",
    "  fig, ax = plt.subplots()\n",
    "  plt.axis([-1.5, 1.5, -1.5, 1.5])\n",
    "  ax.scatter(xs, ys, c='b', marker='s', label=f'[1.0,0] rotated {deg:.2f} degrees')\n",
    "  if Count == -2:\n",
    "    plt.title( vprt(f'Showing [1.0,0] rotated to {deg} degrees. {len(Rtn)} points shown'))\n",
    "    Count += 1\n",
    "  elif Count == -1:\n",
    "    plt.title(vprt(f'Showing [1.0,0] rotated to {deg}. {len(Rtn)} points shown', f'About to show {len(outerIdx)} dot products'))\n",
    "    Count += 1\n",
    "  elif Count < len(outer):\n",
    "    currDot = outerIdx[Count]\n",
    "    currPoints = Rtn[currDot[0]],Rtn[currDot[1]]\n",
    "    cxs = [ t.tolist()[0] for t in currPoints]\n",
    "    cys = [ t.tolist()[1] for t in currPoints]\n",
    "    ax.scatter(xso[0:Count], yso[0:Count], c='r', marker='o', label='prior dot product')\n",
    "    ax.scatter(xso[Count:Count+1], yso[Count:Count+1], c='g', marker='o', label='current dot product')\n",
    "    ax.scatter(cxs, cys, c='y', marker='s', label=f'current points in dot product')\n",
    "    plt.title(vprt(f'Showing [1.0,0] rotated to {deg} degrees. {len(Rtn)} points shown', \n",
    "                   f'showing {Count+1}/{len(outer)} dot product', \n",
    "                   hprt(a=currPoints[0],b=currPoints[1])))\n",
    "    Count += 1\n",
    "    if Faster:\n",
    "      Count = len(outer)\n",
    "  elif Count == len(outer):\n",
    "    # reached end\n",
    "    plt.title(vprt(f'Showing [1.0,0] rotated to {deg} degrees', f'showed all {len(outerIdx)} dot products', \n",
    "                  f'About to show [1,0] rotated to {deg+15} degree'))\n",
    "    ax.scatter(xso[:Count+1], yso[:Count+1], c='r', marker='o', label='dot product')\n",
    "    Rtn.append(m3 @ Rtn[-1])\n",
    "    Count = -2\n",
    "  ax.axhline(y=0, color='k')\n",
    "  ax.axvline(x=0, color='k')\n",
    "  if len(Rtn) <= 3 and not Faster:\n",
    "    plt.legend()\n",
    "  #fig.set_size_inches(4, 4)\n",
    "  plt.axis('equal')\n",
    "  plt.show()\n",
    "  print(f'{len(Rtn)-1=} {len(outer)=} {Count=} {currDot=}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3692d232-a5cd-483d-a820-ef700bab28cc",
   "metadata": {},
   "outputs": [],
   "source": [
    "# run this cell repeatedly!\n",
    "Faster = 1\n",
    "setupDeg()\n",
    "drawDeg()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e49d21b-349d-405e-8704-3fd481438367",
   "metadata": {},
   "source": [
    "### The Hypothesis behind using 2D rotations is:\n",
    " - dot product of 2d rotations preserves some information about where where the original points were\n",
    " - Similarly, subtractions/additions of 2d points preserve some information about the original two points.\n",
    " - Therefore, by perturbing the encoding of the tokens with their relative positions, we hypothesize that the LLM model learns to pick up clues about the spatial positions of the tokens and how they might influence the \"next token\" decision.\n",
    "### We really don't know **WHY** Positional Encoding works.\n",
    "### We also don't know why Rotary PE is better than normal PE.\n",
    "### We also don't understand why we need to perturb different points of the token embedding differently than other parts of the embedding, for the **same** token"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83d42ff3-8393-4da8-a7bc-faed637b2195",
   "metadata": {},
   "source": [
    "### How is it actually coded up in LLMs?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fdfb0ffa-75b4-4460-88cb-f064f6607bb1",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# https://pytorch.org/docs/stable/generated/torch.outer.html\n",
    "v1 = torch.arange(1., 5.)\n",
    "v2 = torch.arange(1., 4.)\n",
    "v3 = torch.outer(v1, v2)\n",
    "v3a = v1.unsqueeze(1) @ v2.unsqueeze(0)\n",
    "# v1 (n) -> (n,1), v2 (m) -> (1,m) --> (n,m)\n",
    "print(hprt(Shapes(v1=v1, v2=v2, v3=v3, v3a=v3a)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1226f1d6-ed9b-4dfc-9570-99425041ef8f",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# https://pytorch.org/docs/stable/generated/torch.polar.html\n",
    "import numpy as np\n",
    "abs = torch.tensor([1, 1, 1, 1], dtype=torch.float64)\n",
    "# note: pi/4 == 45 degrees!\n",
    "angle = torch.tensor([np.pi / 2, np.pi / 4, 5*(np.pi / 4), np.pi/6], dtype=torch.float64)\n",
    "z = torch.polar(abs, angle)\n",
    "print(printTensors(abs=abs, angle=angle, z=z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "73f0e10b-d319-4fff-9745-e00e987675a2",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a97f826a-f6f7-4c20-aa58-a42a7a36bb4f",
   "metadata": {},
   "outputs": [],
   "source": [
    "# https://pytorch.org/docs/stable/generated/torch.ones_like.html\n",
    "# https://pytorch.org/docs/stable/generated/torch.empty.html#torch-empty\n",
    "input = torch.empty(2, 3)\n",
    "output = torch.ones_like(input)\n",
    "print(Shapes(input=input, output=output))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a3c8649f-93b0-47db-8b1e-99cf0784542f",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# https://pytorch.org/docs/stable/generated/torch.view_as_complex.html\n",
    "# https://pytorch.org/docs/stable/generated/torch.view_as_real.html\n",
    "K = torch.arange(16).view(2,2,2,-1).float()\n",
    "C = torch.view_as_complex(K)\n",
    "D = torch.view_as_real(C)\n",
    "print(printTensors(K=K, C=C, D=D))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa12ee45-0f5f-4adf-a593-11b663bb886d",
   "metadata": {},
   "source": [
    "### The following code is adopted from working Llama3\n",
    "  - `precompute_theta_pos_frequencies()` is called before the LLM actually examines any tokens\n",
    "  - `apply_rotary_embedding()` is called from the Attention Layer,\n",
    "  - input tensor `x` is of shape $(B, seqLen, nEmbed)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0f02f70c-e863-4d8b-8dac-ffb116a5ec59",
   "metadata": {},
   "outputs": [],
   "source": [
    "def precompute_theta_pos_frequencies(head_dim: int, seq_len: int, theta: float = 10000.0):\n",
    "  # As written in the paragraph 3.2.2 of the paper\n",
    "  # >> In order to generalize our results in 2D to any xi ∈ Rd where **d is even**, [...]\n",
    "  assert head_dim % 2 == 0, \"Dimension must be divisible by 2\"\n",
    "  # Build the theta parameter\n",
    "  # According to the formula theta_i = 10000^(-2(i-1)/dim) for i = [1, 2, ... dim/2]\n",
    "  # Shape: (Head_Dim / 2)\n",
    "  theta_numerator = torch.arange(0, head_dim, 2).float()\n",
    "  # Shape: (Head_Dim / 2)\n",
    "  theta = 1.0 / (theta ** (theta_numerator / head_dim)) # (Dim / 2)\n",
    "  # Construct the positions (the \"m\" parameter)\n",
    "  # Shape: (Seq_Len)\n",
    "  m = torch.arange(seq_len)\n",
    "  # Multiply each theta by each position using the outer product.\n",
    "  # Shape: (Seq_Len) outer_product* (Head_Dim / 2) -> (Seq_Len, Head_Dim / 2)\n",
    "  freqs = torch.outer(m, theta).float()\n",
    "  # We can compute complex numbers in the polar form c = R * exp(m * theta), where R = 1 as follows:\n",
    "  # (Seq_Len, Head_Dim / 2) -> (Seq_Len, Head_Dim / 2)\n",
    "  freqs_complex = torch.polar(torch.ones_like(freqs), freqs)\n",
    "  return freqs_complex, freqs, m, theta, theta_numerator\n",
    "\n",
    "def apply_rotary_embeddings(x: torch.Tensor, freqs_complex: torch.Tensor):\n",
    "  # Separate the last dimension pairs of two values, representing the real and imaginary parts of the complex number\n",
    "  # Two consecutive values will become a single complex number\n",
    "  # (B, Seq_Len, H, Head_Dim) -> (B, Seq_Len, H, Head_Dim/2)\n",
    "  x_complex = torch.view_as_complex(x.float().reshape(*x.shape[:-1], -1, 2))\n",
    "  # Reshape the freqs_complex tensor to match the shape of the x_complex tensor. \n",
    "  # So we need to add the batch dimension and the head dimension\n",
    "  # (Seq_Len, Head_Dim/2) --> (1, Seq_Len, 1, Head_Dim/2)\n",
    "  freqs_complex = freqs_complex.unsqueeze(0).unsqueeze(2)\n",
    "  # Multiply each complex number in the x_complex tensor by the corresponding complex number in the freqs_complex tensor\n",
    "  # Which results in the rotation of the complex number as shown in the Figure 1 of the paper\n",
    "  # (B, Seq_Len, H, Head_Dim/2) * (1, Seq_Len, 1, Head_Dim/2) = (B, Seq_Len, H, Head_Dim/2)\n",
    "  print(f'{list(x_complex.shape)} @ {freqs_complex.shape}')\n",
    "  x_rotated = x_complex * freqs_complex\n",
    "  # Convert the complex number back to the real number\n",
    "  # (B, Seq_Len, H, Head_Dim/2) -> (B, Seq_Len, H, Head_Dim/2, 2)\n",
    "  x_out = torch.view_as_real(x_rotated)\n",
    "  # (B, Seq_Len, H, Head_Dim/2, 2) -> (B, Seq_Len, H, Head_Dim)\n",
    "  x_out = x_out.reshape(*x.shape)\n",
    "  return x_out.type_as(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5c3ef01b-3afb-491b-b1d3-135e4c5225d6",
   "metadata": {},
   "outputs": [],
   "source": [
    "nEmbed = 8\n",
    "seqLen = 6\n",
    "nHeads = 1\n",
    "nBatch = 1\n",
    "headDim = nEmbed // nHeads "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b0c3b5be-240c-4907-a8b7-d01006304226",
   "metadata": {},
   "outputs": [],
   "source": [
    "PreComputed = precompute_theta_pos_frequencies(headDim, seqLen)\n",
    "freqs_complex, freqs, m, theta, theta_numerator = PreComputed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1d1b3e4b-b116-479d-ab62-e438d7db921e",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "print(vprt(Shapes(theta_numerator=theta_numerator), Shapes(theta=theta), Shapes(m=m), Shapes(freqs=freqs), Shapes(freqsC=freqs_complex)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e19c148d-3ef1-433f-8b3d-24b8ec3ea612",
   "metadata": {},
   "outputs": [],
   "source": [
    "print(printTensorsAll(theta_numerator=theta_numerator))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3484b306-8683-4fd9-9700-e7b04139fb75",
   "metadata": {},
   "source": [
    "# x0 is a fake embedding vector for some token, with special values to see what effect RPE has"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f4185f87-b2e1-4cea-8ba5-e1badc012b92",
   "metadata": {},
   "outputs": [],
   "source": [
    "x00 = torch.tensor([1, 0], dtype=torch.float)\n",
    "x0 = torch.cat([x00, x00, x00, x00])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a2d4f2d4-b067-45f4-bb43-a0781c5e0b4b",
   "metadata": {},
   "outputs": [],
   "source": [
    "print(printTensors(x0=x0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "15937d80-2c1b-42b2-abb9-e8866f7bc872",
   "metadata": {},
   "outputs": [],
   "source": [
    "# (B, seqLen, H, headDim)\n",
    "# or (1, 6, 1, 8)\n",
    "x = torch.stack([x0]*seqLen).unsqueeze(0).unsqueeze(2)\n",
    "print(printTensors(x=x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f58ca836-efcc-42f2-9fc1-4b57758c4bef",
   "metadata": {},
   "source": [
    "### The above stack and unsqueeze is needed because the above  apply_rotary_embeddings() is found in the attention layer.\n",
    "  - need to take into account $(B, seqLen, H, HeadDim)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "698c7f81-763d-433b-aeaf-0c87c7bbe779",
   "metadata": {},
   "outputs": [],
   "source": [
    "x.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d0f10668-3f86-4237-86e7-d9c35df70415",
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = x.float().reshape(*x.shape[:-1], -1, 2)\n",
    "x_complex = torch.view_as_complex(x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cf11268a-5636-4360-9422-1a783adeb1c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "print(printTensors(x0=x0.view(-1,2), x=x, x1=x1, x_complex=x_complex))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "62155381-a5d0-4c1d-a281-9e532a34e411",
   "metadata": {},
   "outputs": [],
   "source": [
    "f2 = freqs_complex.unsqueeze(0).unsqueeze(2)\n",
    "print(vprt(printTensors(freqs_complex=freqs_complex), printTensors(f2=f2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b29932c4-fb2b-484f-9ace-9e3d2443959a",
   "metadata": {},
   "outputs": [],
   "source": [
    "y = apply_rotary_embeddings(x, freqs_complex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4ec5eccc-a881-4c12-bde6-dbe856a79db7",
   "metadata": {},
   "outputs": [],
   "source": [
    "print(hprt(y=y.squeeze(), x=x.squeeze()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "87ab337c-8ca6-4adf-957f-12c2694a0d60",
   "metadata": {},
   "outputs": [],
   "source": [
    "y1 = y.squeeze()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0d7233f7-7f37-4f6c-87d1-7eff23a36500",
   "metadata": {},
   "outputs": [],
   "source": [
    "PT = 0\n",
    "r = 0\n",
    "def drawRPE(y, x0):\n",
    "  global fig, ax, PT, r\n",
    "  seqLen,nEmbed = y.shape\n",
    "  assert nEmbed == x0.shape[0]   \n",
    "  assert nEmbed % 2 == 0\n",
    "  nPoints = nEmbed // 2\n",
    "  if r >= seqLen:\n",
    "    r = 0\n",
    "    PT = 0\n",
    "    return\n",
    "  # for each seqLen\n",
    "  orig = x0.view(-1,2)\n",
    "  repl = y[r].view(-1,2)\n",
    "  fig, ax = plt.subplots()\n",
    "  plt.axis([-1.5, 1.5, -1.5, 1.5])\n",
    "  ax.axhline(y=0, color='k')\n",
    "  ax.axvline(x=0, color='k')\n",
    "  if PT < nPoints:\n",
    "    origPT = orig[PT]\n",
    "    replPT = repl[PT]\n",
    "    oxs = orig[:, 0]\n",
    "    oys = orig[:, 1]\n",
    "    rxs = repl[:, 0]\n",
    "    rys = repl[:, 1]\n",
    "    assert origPT[0] == oxs[PT] and origPT[1] == oys[PT]\n",
    "    plt.title(hprt(f'row {r}, point {PT}', ORIG=origPT, REPL=replPT))\n",
    "    ax.scatter(oxs[PT], oys[PT], c='r', marker='o', label='original')\n",
    "    ax.scatter(rxs[PT], rys[PT], c='g', marker='s', label='rotated')\n",
    "    PT += 1\n",
    "  else:\n",
    "    plt.title(f'showed row {r}. abouto to show row {r+1}')\n",
    "    r += 1\n",
    "    PT = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8287f365-fa58-4780-85ab-99ca45c9121e",
   "metadata": {},
   "outputs": [],
   "source": [
    "drawRPE(y1, x0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2425da69-4f57-4903-a4a6-2a18b6265b76",
   "metadata": {},
   "outputs": [],
   "source": [
    "nPoints = 360 // 15\n",
    "nEmbed = 2 * nPoints\n",
    "seqLen = 8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "09750049-cfe5-4bb3-a770-45d1eb798e7c",
   "metadata": {},
   "outputs": [],
   "source": [
    "nHeads = 1\n",
    "nBatch = 1\n",
    "headDim = nEmbed // nHeads \n",
    "# using only one head!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6df44744-fd32-42ea-90b3-9754d28fdff9",
   "metadata": {},
   "outputs": [],
   "source": [
    "original = torch.tensor([1,0]).double()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fa7093de-8a2a-43a5-bad7-46c1a5ae23f4",
   "metadata": {},
   "outputs": [],
   "source": [
    "original.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4ac1f579-34d1-45e3-98e0-665b300a293a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# x0 = torch.stack([ rot2d(toRAD(15*k)) @ original for k in range(nPoints) ]).float()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d209672f-68b8-4962-8b44-7042bfc5095c",
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = torch.stack( [original for k in range(nPoints) ]).float()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a86cddaa-2c43-473a-b6a5-4e95808aec39",
   "metadata": {},
   "outputs": [],
   "source": [
    "x0.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5fb8acdb-3b33-4164-8e5d-d3645f0e4f5f",
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = torch.stack([x0.view(-1)]*seqLen).unsqueeze(0).unsqueeze(2)\n",
    "x1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cfd8bd45-0200-4262-8a29-e60f9ce879dd",
   "metadata": {},
   "outputs": [],
   "source": [
    "x2 = x1.unsqueeze(0).unsqueeze(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cb49df9d-b02e-4cfe-a50c-f01f2bacf2f7",
   "metadata": {},
   "outputs": [],
   "source": [
    "x2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "26bc1526-ed0e-4b26-8c61-158aed2bb107",
   "metadata": {},
   "outputs": [],
   "source": [
    "PreComputed = precompute_theta_pos_frequencies(headDim, seqLen)\n",
    "freqs_complex, freqs, m, theta, theta_numerator = PreComputed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "18c6c069-fa83-4299-bccb-dedeb99905ae",
   "metadata": {},
   "outputs": [],
   "source": [
    "y = apply_rotary_embeddings(x2, freqs_complex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1c052b57-8130-4b61-8962-612cd9ee60e5",
   "metadata": {},
   "outputs": [],
   "source": [
    "y1 = y.squeeze()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0463199e-ae8b-4a74-97a0-f8c0f78a026e",
   "metadata": {},
   "outputs": [],
   "source": [
    "PT = 0\n",
    "r = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1e1cf4fa-3631-4f08-af83-814e468763c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "x0.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "927aa051-7ad7-404c-b5e4-52a30b9ee120",
   "metadata": {},
   "outputs": [],
   "source": [
    "drawRPE(y1, x0.view(-1))"
   ]
  }
 ],
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