@article {4375,
	title = {Double descent in the condition number},
	year = {2019},
	month = {12/2019},
	abstract = {<p>In solving a system of n linear equations in d variables\&nbsp;\&nbsp; Ax=b, the condition number of the (n,d) matrix A measures how\&nbsp;\&nbsp; much errors in the data b affect the solution x. Bounds of\&nbsp;\&nbsp; this type are important in many inverse problems. An example is\&nbsp;\&nbsp; machine learning where the key task is to estimate an underlying\&nbsp;\&nbsp; function from a set of measurements at random points in a high\&nbsp;\&nbsp; dimensional space and where low sensitivity to error in the data is\&nbsp;\&nbsp; a requirement for good predictive performance. Here we report the\&nbsp;\&nbsp; simple observation that when the columns of A are random vectors,\&nbsp;\&nbsp; the condition number of A is highest, that is worse, when d=n,\&nbsp;\&nbsp; that is when the inverse of A exists. An overdetermined system\&nbsp;\&nbsp; (n\&gt;d) and especially an underdetermined system (n\&lt;d), for which\&nbsp;\&nbsp; the pseudoinverse must be used instead of the inverse, typically\&nbsp;\&nbsp; have significantly better, that is lower, condition numbers. Thus\&nbsp;\&nbsp; the condition number of A plotted as function of d shows a\&nbsp;\&nbsp; double descent behavior with a peak at d=n.</p>
},
	author = {Tomaso Poggio and Gil Kur and Andrzej Banburski}
}
