In 1weintroduce ourmotivations, ourmain results and discuss the related work.

In previous work [12, 13], a method was introduced that leverages compressive sensing techniques tofind thefewest taxa thatfitsthefrequencyofshort sequences ofnucleotides (i.e., k-mers) in a given sample. Consider, for instance, an environment/sample made of s bacterial species but where two of them are almost identical: one would wish to say that the concentration vector is almost(s 1)-sparse rather thans-sparse!

When n = 2, the constraint matrixA has E = I2 1>2 and G = 1>2 I2. Now we simplify the matrixAby removing a specific set of redundantrows. Furthermore, the rows of A are categorized into a single set so that the criterion in Proposition 3.2 holds true (thedashed lineintheformulation of Aservesasapartition ofthissingle setintotwosets). We use the proof by contradiction. In particular, assume that problem(10) is a MCF problem whenm 3andn 3,Proposition 3.3 implies that the constraint matrixAisTU.

We consider a target neural networkf: Rd0 Rdl of depth l, which is described as follows. Similar to the previous works [6, 7], we assume that g(x) is twice as deep as the target network f(x). Thus, g(x) can be described as g(x)=G2lσ(G2l 1σ( G1(x))), (2) where Gj is a edj edj 1 matrix (edj N 1 for j = 1,,2l) with ed2i = di. Under this re-sampling assumption, we describe our main theorem as follows. 1 Theorem A.1 (Main Theorem) Fix,δ>0, and we assume thatkFikFrob 1. LetR Nand we assumethat each elementof Gi can be re-sampled with replacementfrom the uniformdistribution U[ 1,1] up to R 1 times. If n 2log(1δ) holds, then with probability at least 1 δ, we have |α Xi|, (5) for some i {1,,n}.