Quoted from section 4.1 in the paper Aggregating Crowdsourced Binary Ratings:

... We will also denote the (scaled) top eigenvector of a matrix $M$ as $v_1(M) = \arg\min_x \|M - xx^T\|_2, x^{(1)}\ge0$.

I understand the concept of scaled leading eigenvector, but I don't understand the constraint $x^{(1)}\ge0$. Specifically, what is the meaning of the superscript $(1)$?

My first guess was that it means the first entry of $x$, but in the rest of the paper, they used the standard notation $x_i$ to represent vector entries, so I don't think this is the case.

Edit: Rodrigo de Azevedo suggests in the comments that different co-authors of the paper might use different notations, and perhaps it does mean $x_1\ge0$. If this is the case, I have the following questions:

  • Have you seen any paper (in applied mathematics and related fields) that uses this weird notation?
  • What is the reason behind the $x_1\ge0$ requirement? Can we somehow relate the $v_1(M)$ in this definition with the leading eigenvector in the "standard" definition $\arg\max_x \frac{x^TMx}{\|x\|}$?
  • $\begingroup$ Have you tried contacting the authors? $\endgroup$ – Rodrigo de Azevedo Jan 4 at 9:17
  • $\begingroup$ @RodrigodeAzevedo Not yet, but I've found a few typos and unrigorous proofs in the paper, so I guess it's some kind of mistake :( $\endgroup$ – hklel Jan 4 at 21:09
  • $\begingroup$ Four authors. Perhaps one started using $x^{(1)}$ to denote the first entry of vector $x$, and then some other decided to use $x_1$ instead. And they forgot to change the notation in one instance. Perhaps. $\endgroup$ – Rodrigo de Azevedo Jan 4 at 21:23
  • $\begingroup$ @RodrigodeAzevedo Thanks for the reply! If that's the case, can you take a look at the updated question? $\endgroup$ – hklel Jan 4 at 22:03

My guess is that it is the first component of the vector. This is so that the problem has a more unique solution since $xx^T = (-x)(-x)^T$. This can still fail if $x^{(1)}=0$ but depending on the matrices involved this could be very unlikely


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