Home Knowledge Base \mathbb{R}^m$

\mathbb{R}^m$

No mentions found

This entity hasn't been tracked yet, or Iris is still building its knowledge base.

Related Articles from SNS

$\mathbb{R}^{2k}$ is Theoretically Large Enough for Embedding-based Top-$k$ Retrieval

Announce Type: replace Abstract: This paper studies the Minimal Embeddable Dimension (MED): the least dimension in which there exists a configuration of $m$ object vectors so that every subset of size at most $k$ is exactly retrieved by score comparison. Our result shows MED is $\Theta(k)$, independent of $m$, for inner product, Euclidean distance, and cosine similarity. We then consider Robust MED (RMED), where all vectors are unit normed and an $\epsilon$ gap of scores is required.

arXiv CS 7d ago

Ordinals and recursively defined functions on the reals

arXiv:2311.17210v5 Announce Type: replace-cross Abstract: We determine sufficient conditions under which certain recursively defined functions are well defined for all real inputs. Given a function $f:\mathbb R\to\mathbb R$, call a decreasing sequence $x_1>x_2>x_3>\cdots$ "$f$-bad" if $f(x_1)>f(x_2)>f(x_3)>\cdots$, and call the function $f$ "ordinal decreasing" if there exist no infinite $f$-bad sequences. We prove the following result:

arXiv CS 1d ago

BSTabDiff: Block-Subunit Diffusion Priors for High-Dimensional Tabular Data Generation

Announce Type: new Abstract: High-Dimensional Low-Sample Size (HDLSS) tabular domains (e.g., omics) are characterized by $n \ll m$, where $n$ = number of samples, and $m$ = number of features. Such domains often exhibit strong local correlation groups, sparse cross-group dependencies, heavy-tailed non-Gaussian marginals, heteroscedastic noise, and structured missingness, making direct density learning in $\mathbb{R}^m$ ill-conditioned since $n \ll m$. We propose BSTabDiff, a block-subunit...

arXiv CS 1d ago

Almost covering all the layers of hypercube with multiplicities

Announce Type: replace-cross Abstract: Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of...

arXiv CS 2d ago

Collision Resistance of Single-Layer Neural Nets

arXiv:2606.03807v1 Announce Type: new Abstract: We initiate the study of the algorithmic complexity of finding collisions in single-layer binary neural networks. Given a random matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$, an input $\mathbf{x} \in \{-1,1\}^n$ is mapped to a binary output vector $\varphi(\mathbf{A}\mathbf{x})\in \{-1,1\}^m$, where $\varphi$ is an activation function with constant behavior on $[\kappa, \infty)$ for some threshold $\kappa \geq 0$. We identify the threshold...

arXiv CS 7d ago

Ultrafast machine learning on FPGAs via Kolmogorov-Arnold Networks

Ultrafast machine learning on FPGAs via Kolmogorov-Arnold Networks This post is a high-level explainer for my Master’s thesis, which involves designing hardware architectures for ultrafast inference and online learning using the Kolmogorov-Arnold Network (KAN) architecture. I’ll assume familiarity with standard machine learning concepts, as well as some understanding of hardware and digital circuits; read my previous post here for the latter. Please read the two papers below for more...

Hacker News 1d ago

Deep learning four decades of human migration

Abstract Human migration is a fundamental driver of global demographic change, shaping population structure, labour markets and social policy across countries1,2,3. Although long-term migration patterns are often linked to economic development4, they can shift rapidly in response to shocks such as conflict, environmental crises and political change5. Despite its importance, migration remains difficult to measure consistently: existing data are sparse, concentrated in high-income settings and...

Nature 21h ago

A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space

arXiv:2606.07325v1 Announce Type: cross Abstract: We study the minimax rate of estimating a future value $\mu_{t_n+h}$ of a curve $t\mapsto\mu_t$ in the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\|\nabla_t^k v\|\le\varepsilon$ on the $k$-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every...

arXiv CS 2d ago

Scale-Invariant Neural Network Optimization: Norm Geometry and Heavy-Tailed Noise

arXiv:2605.18528v2 Announce Type: replace-cross Abstract: A growing lesson from neural network optimization is that optimizer design should respect how the model is parametrized. Scale-invariant methods become important because their normalized layerwise updates can not only support hyperparameter transfer across model sizes but exploit input-output matrix norm geometry. At the same time, stochastic gradient noises in deep learning are often far from sub-Gaussian and may exhibit heavy tails.

arXiv CS 8d ago

Magenta RealTime 2: Open and Local Live Music Models

We’re excited to share Magenta RealTime 2 (MRT2), a state-of-the-art open model and efficient real-time inference engine that enables you to build and play AI musical instruments on your laptop! To get started, download the apps on your MacBook (requires Apple Silicon). Unlike other large generative music models that work offline to turn a prompt into a track, MRT2 is a live, interactive model that you can control with MIDI and audio, in addition to text.

Hacker News 5d ago