Why Your Brain Matters in the Era of AI: You Can’t Outsource Intuition

Over the past few years, LLMs have developed rapidly, significantly expanding our cognitive capabilities. Many people probably now turn to AI first when they encounter a problem. It may be that LLMs already know more than my brain does, and perhaps possess higher reasoning abilities than my brain. However, in this article, I will discuss […]

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Sorting with LLMs

Sorting is a classic task in computer science, but it has recently intersected with state-of-the-art LLMs and sparked a new research trend. Sorting can be executed as long as you define a comparison function. Traditional comparison functions assumed measurable numeric quantities such as height, price, or distance. However, if we call an LLM inside the

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Speculative Decoding Explained

“Generating…”We are all familiar with the cursor blinking, slowly revealing text word by word. Whether it’s ChatGPT or a local LLM, the generation speed of autoregressive models based on Transformers is fundamentally bound by their sequential nature. It computes, emits one token, computes again, emits the next token, and repeats. This strict dependence chain introduces

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Even GPT-5.2 Can’t Count to Five

In this post, we discuss how state-of-the-art large language models still make mistakes on extremely simple problems based on Even GPT-5.2 Can’t Count to Five: The Case for Zero-Error Horizons in Trustworthy LLMs. For concrete examples: if you ask whether the number of 1s in 11000 is even or odd, gpt-5.2-2025-12-11 answers “odd”. If you

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How LLMs Really Do Arithmetic

LLMs can answer prompts like “226-68=” by outputting “158”, but it turns out that this computation is carried out in a much stranger way than we might imagine, as shown by [Nikankin+ ICLR 2025]. Let us first confirm the assumptions. We do not use chain-of-thought. We consider the setting where the model directly outputs an

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TwoNN Intrinsic Dimension Explained: Python and Visual Illustrations

Real-world data often live in a high-dimensional ambient space, but the points themselves concentrate near a much lower-dimensional manifold. The “visible” (ambient) dimension is easy to read off from the feature vector length, while the “intrinsic” dimension (the effective degrees of freedom) is much harder to estimate. Two Nearest Neighbors (TwoNN) is a simple yet

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Fisher Information Explained: Python and Visual Illustrations

Definition of Fisher Information The Fisher information is defined as $$\mathrm{FisherInformation}(\theta_0)\stackrel{\text{def}}{=}-\mathbb{E}_{X\sim p(x\mid\theta_0)}\left[\frac{d^2}{d\theta^2}\log p(x\mid\theta)\bigg|_{\theta=\theta_0}\right].$$ Fisher information quantifies how precisely a model parameter can be estimated.A larger Fisher information means the parameter can be estimated more accurately,while a smaller Fisher information indicates that estimation is more difficult. Fisher information admits several equivalent interpretations. Equivalent Expressions $$\begin{align}&\mathrm{FisherInformation}(\theta_0) \\&\stackrel{\text{def}}{=}-\mathbb{E}_{X

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