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Polynomial Maps from $S^n$ to $S^n$

Motivated by https://x.com/gabrielpeyre/status/1837156819799577034

---

Understanding Polynomial Maps on Spheres

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Variant of AM-GM for Minimization

When dealing with functions of the form $f(x) = x^a + \frac{1}{x^b}$, a variant of the AM-GM inequality can be used to find the minimum. Specifically, if you have:

$$ f(x) = c_1 \cdot x^a + c_2 \cdot \frac{1}{x^b} $$

The minimum occurs at:

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Summarization:

As a professional summarizer, create a concise and comprehensive summary of the provided text, be it an article, post, conversation, or passage, while adhering to these guidelines:

Craft a summary that is detailed, thorough, in-depth, and complex, while maintaining clarity and conciseness.

Incorporate main ideas and essential information, eliminating extraneous language and focusing on critical aspects.

Rely strictly on the provided text, without including external information.

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def plot_lr_final_loss_batchsize(file_path):
    # Load the data
    data = pd.read_csv(file_path)
    
    # Extract columns that match 'val_loss/val_loss_'
    val_loss_columns = [col for col in data.columns if col.startswith('val_loss/val_loss_')]
    
    # Sort val_loss_columns by K (numeric value after 'val_loss/val_loss_') in increasing order

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import os
import torch
import json
from PIL import Image
from torch.utils.data import Dataset
from diffusers.models import AutoencoderKL
from streaming import MDSWriter

import logging
import time

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import os
import torch
import json
from PIL import Image
from torch.utils.data import Dataset, DataLoader
from diffusers.models import AutoencoderKL
from streaming import MDSWriter

import logging
import time

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# Suppose you have neural network that
# x_l = a_l * W_l x_{l-1}, W_l_{i,j} ~ N(0, b_l^2), Learning rate of W_l := c_l,
# If you are using adam, you can
# a_l <- a_l * A , b_l <- b_l / A, c_l <- c_l / A
# and it will have exactly identical training dynamics as before.
# This is known as ABC (ABCD) redundancy. For more general case: https://arxiv.org/abs/2308.01814
# Let me show you what I mean:


import torch

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import requests
import json
import base64
from PIL import Image
from io import BytesIO
import os
import openai
from openai import OpenAI

import fal_client

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#### Inference utils
import math

import torch
import torch.nn as nn
import torch.nn.functional as F

from transformers import AutoTokenizer, AutoModelForSeq2SeqLM
from diffusers.image_processor import VaeImageProcessor
from diffusers.models import AutoencoderKL

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[
    {
        "description": "a flamboyant octopus dressed in a tuxedo is hosting an undersea tea party on a coral reef, complete with tiny teacups and saucers made from seashells. Each of the octopus's eight arms is elegantly balancing a different teapot, pouring colorful and sparkly liquid into the cups. Surrounding the octopus are a variety of sea creatures dressed in formal wear: a well-dressed clownfish is toasting with a miniature glass, a sea turtle wears a monocle and is engaged in deep conversation with a stylishly-attired seahorse, and a group of jellyfish float nearby, their translucent bodies glowing softly in the ambient light. Coral formations around them resemble extravagant table decorations, draped in strands of pearls and glittering seaweed, while a vibrant school of fish swims by, seemingly mesmerized by the spectacle. The scene is filled with a sense of whimsy and elegance, illuminated by streams of sunlight filtering down from above, creating a magical underwater atmosphere.",
        "