Order-optimal Compressive Sensing for Approximately k -sparse Signals:

1. Order-optimal Compressive Sensing for Approximately k-sparse Signals: O(k) measurements and O(k) decoding steps MayankBakshi, SidharthJaggi, Sheng Cai, Minghua Chen Overview Exactly - sparse A- sparse Measurement operation: Unknowns: Signal “Noise” Objective: Design , decoder s.t. estimation error “small”, i.e., w.h.p. Previous design fails: Measurement design: - “small” noise in => possibly large noise in phase of and => identification/verification error - Estimation error propagates (and amplifies) over iterations identification Figure 5 verification Three new ideas: 1. Truncation: Figure 8 - Noise does not change phase much Our Result: a.measurements suffice b. “SHO(rt)-FA(st)” algorithm: steps suffice c. processed “bits”/operations - Most of the norm of captured 2. Repeated identification/verification measurements - Represent each node on left as a sequence of digits - Separate identification/verification measurements for each digit SHO-FAdecoder: Information Theoretically order-optimal Settings: a.Exactly -sparse b. Approximately k-sparse with for some . Figure1 Figure 6 at all right nodes; Check if leaf steps Figure 9 …… 1st digit 3rd digit 3. Concatenation and Coupon Collection ? # outputs = ? Y Declare success Y Key tool: “Almost” Expanders Key tool: “Almost” Expanders Construction of graph : N N Declare failure Pick deg=3 Figure 2 At most iterations Figure 10 - Bipartite, left regular - uniformly chosen neighbours of each left node a. Identify signal coordinate, • s.t. • b. Output 1 …... …... 1 2 Subtract contribution of from 2 …... 3 ck 3 n at each neighbour of ; Update Check if leaf 4 …... 4 5 1. High probability of vertex expansion: - Every set S of size at most k (and all its subsets) have expansion at least with a high probability over the construction of - Run SHO-FA independently on chunks, each of size , recover most of the signal Check if leaf Figure 7 Input: Node for some ? • Reconstruct the failed locations by looking for leafs in random linear combinations • - like coupon collection Output: Is a leaf? N is not a leaf Figure 3 S :support of Identify References Expands Does not expand [1] Accompanying short writeup available at http://personal.ie.cuhk.edu.hk/~mayank/CS/writeup.pdf [2] M. Bakshi, S. Jaggi, S. Cai, M. Chen, “Order-optimal compressive sensing for k-sparse signals with noisy tails: O(k) measurements, O(k) steps”, pre-printavailable at http://personal.ie.cuhk.edu.hk/~sjaggi/CS_)1.pdf, Video at http://youtu.be/UrTsZX7-fhI 2. “Many” S-leaf nodes Y ? N is not a leaf Figure 4 Verify |S| ≥2|S| Y is a leaf