For an $m\times n$ matrix $A$ of rank $r \ll \min(m, n)$. $A$ has $r$ positive singular values: $$ 0\le \sigma_r \le \ldots \le \sigma_2 \le \sigma_1. $$ Denote as $\sigma\in \R^r$ the singular value vector. Its norms would satisfy: Spectral norm $$ \Vert A\Vert _2 = \sigma_1=\Vert \sigma\Vert _{\infty}. $$ Frobenius norm $$ \Vert A \Vert_F = \sqrt{\sum_{i=1}^r \sigma_i^r} = \Vert \sigma\Vert_2 $$ Nuclear norm $$ \Vert A\Vert_* = \sum_{i =1 }^r \sigma_i = \Vert \sigma\Vert_1. $$ $\Vert A\Vert_F $’s order will be $O(\sqrt{r} \sigma_{\max})$, and $\Vert A\Vert_*$ will be of $O(r \sigma_{\max})$. Actually ...

Whenever I look up matrix norms, I almost inevitably end up on Wikipedia. It’s usually the most reliable reference. Yet I also often find myself baffled by sentences like: “In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von Neumann norm) arises as a generalization of $p$-integrability similar to the trace class norm and the Hilbert–Schmidt norm.” ?!? This is where the recursive study begins: confused by one definition, I open another page to clarify it, only to be met with two more unfamiliar terms, and on and on. This post is an attempt to stop the recursion. Here’s a simplified, self-contained guide to matrix norms. ...

During dinner my friends debated about what kind of egg one should buy. Some says that big (XL/jumbo) eggs is painful for hens. People should buy small eggs. One of the first-year student speaks up If everyone goes to buy small eggs, producers do notice that large eggs don’t sell well. Then hens that lay large eggs will lose their commercial value. Farmers will kill them and replace them with hens that lay small eggs. By buying large eggs yourself, you are protecting the jobs and the lives of hens that lay large eggs. ...

In the world of musical instruments and wine, quality is more subjective, and price signals more than just quality. It’s for sure not quasi-linear utility. Sometimes, the more expensive, the merrier. Soloist evaluations of six Old Italian and six new violins Link: https://www.pnas.org/doi/10.1073/pnas.1323367111 10 renowned soloists each blind-tested six Old Italian violins (including five by Stradivari) and six new during two 75-min sessions—the first in a rehearsal room, the second in a 300-seat concert hall. When asked to choose a violin to replace their own for a hypothetical concert tour, 6 of the 10 soloists chose a new instrument. A single new violin was easily the most-preferred of the 12. On average, soloists rated their favorite new violins more highly than their favorite old for playability, articulation, and projection, and at least equal to old in terms of timbre. Soloists failed to distinguish new from old at better than chance levels. ...

Partita is a Baroque-era instrumental composition, essentially a suite or collection of stylized dance movements, often starting with a prelude and including dances like the Allemande (German dance in 4/4), Courante (lively French/Italian in triple meter), Sarabande (A slow, dignified Spanish dance in triple meter), and Gigue (A fast, energetic dance of English/Irish origin). These form a coherent whole for a solo instrument or small group, most famously seen in works by J.S. Bach. ...

“I’m in a mood for rite of spring”, I said “Go check Klaus Mäkelä’s Carnegie hall version. Let me show you”. My friend opened medici for me. FYI, the OG ballet of Rite of Spring depicts various primitive rituals celebrating the advent of spring (in Russia), after which a young girl is chosen as a sacrificial victim and dances herself to death. Klaus carried Rite of Spring to its extreme. His version embodies a raw, primitive connection with the nature, and the ultimate unrestness of human heart. Energy, passion, MADNESS! ...

PaperMod automatically loads any CSS file inside assets/css/extended/. Step 1: CSS Plugin So in order to configure any plug in, just create file assets/css/extended/video-embed.css in Hugo Papermod working directory, and paste .video-embed { position: relative; padding-bottom: 56.25%; /* 16:9 */ height: 0; overflow: hidden; max-width: 800px; margin: 2rem auto; border-radius: 12px; box-shadow: 0 4px 12px rgba(0,0,0,0.15); } .video-embed iframe { position: absolute; inset: 0; width: 100%; height: 100%; border: 0; border-radius: 12px; } @media (prefers-color-scheme: dark) { .video-embed { box-shadow: 0 4px 12px rgba(0,0,0,0.6); } } Step 2: html configure short code Create layouts/shortcodes/video.html paste in: ...

Alla marcia, “in the style of a march”: Emil Gilels played this prelude at a front in World War II, in support for the Soviet militaryforces fighting in the war. The narrator says (in Russian): “Gilels is playing at the front, to remind us what the war is worth fighting for: Immortal music!” Wikipedia The prelude is the most performed and recorded piece of Rachmaninoff’s Op. 23 prelude set. Of all the versions I’ve heard (believe me, a lot of them), Ashkenazy’s version is coarse, rough, resilient, and strong. In my pov, though classical performers are somewhat free to interpret any piece written down, Ashkenazy’s way is the right way that it should be played.

A little reflection to Lu Ming’s speech at China Econ Annual Conference. Now the Chernobyl (the HBO miniseries) becomes relatable: At the trial of deciding who is responsible for the explosion, the scientist Legasov, instead of taking it easy by blaming the explosion to individual’s mistake, tried to point out the real flaw, which faces warning and threat from the official: [Judge] Professor Legasov, if you mean to suggest the ussr is somehow responsible for what happened then I must warn you that you are treading on dangerous ground. ...

Thanks to the internet, I’ve got to follow up with the latest heated discussion at China Econ Annual Conference. During the panel discussion, Professor Lu Ming expressed his concern for Chinese economic ecology: OG question (rephrased): Q1 how to balance academic research and policy research? Q2 how should young scholars plan their career path to do research that benefit society TL;DR: Q1 Economics research (in China) prioritizes methodological sophistication and journal publication over real-world relevance and policy impact. Journals’ incentives have distorted research toward complex but hollow models, misleading young scholars and stripping away China-specific realities. As a result, much academic work is disconnected from actual policy, sometimes even prescribing more regulation to fix problems created by regulation itself. ...