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# Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices

Contributed by David L. Donoho, November 12, 2012 (sent for review August 9, 2012)

### Related Articles

- Toward deterministic compressed sensing- Jan 14, 2013

## Abstract

In compressed sensing, one takes samples of an *N*-dimensional vector using an matrix *A*, obtaining undersampled measurements . For random matrices with independent standard Gaussian entries, it is known that, when is *k*-sparse, there is a precisely determined phase transition: for a certain region in the (,)-phase diagram, convex optimization typically finds the sparsest solution, whereas outside that region, it typically fails. It has been shown empirically that the same property—with the same phase transition location—holds for a wide range of non-Gaussian random matrix ensembles. We report extensive experiments showing that the Gaussian phase transition also describes numerous deterministic matrices, including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Namely, for each of these deterministic matrices in turn, for a typical *k*-sparse object, we observe that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices. Our experiments considered coefficients constrained to for four different sets , and the results establish our finding for each of the four associated phase transitions.

- sparse recovery
- universality in random matrix theory equiangular tight frames
- restricted isometry property
- coherence

Compressed sensing aims to recover a sparse vector from indirect measurements with , and therefore, the system of equations is underdetermined. Nevertheless, it has been shown that, under conditions on the sparsity of , by using a random measurement matrix *A* with Gaussian i.i.d entries and a nonlinear reconstruction technique based on convex optimization, one can, with high probability, exactly recover (1, 2). The cleanest expression of this phenomenon is visible in the large asymptotic regime. We suppose that the object is *k*-sparse—has, at most, *k* nonzero entries—and consider the situation where and . Fig. 1*A* depicts the phase diagram and a curve separating a success phase from a failure phase. Namely, if , then with overwhelming probability for large *N*, convex optimization will recover exactly; however, if , then with overwhelming probability for large *N* convex optimization will fail. [Indeed, Fig. 1 depicts four curves of this kind for —one for each of the different types of assumptions that we can make about the entries of (details below).]

How special are Gaussian matrices to the above results? It was shown, first empirically in ref. 3 and recently, theoretically in ref. 4, that a wide range of random matrix ensembles exhibits precisely the same behavior, by which we mean the same phenomenon of separation into success and failure phases with the same phase boundary. Such universality, if exhibited by deterministic matrices, could be very important, because certain matrices, based on fast Fourier and fast Hadamard transforms, lead to fast and practical iterative reconstruction algorithms, even in the very large *N* setting, where these results would have greatest impact. In certain fast algorithms like FISTA (5) and AMP (6), such matrices are simply applied implicitly and never need to be stored explicitly, saving space and memory accesses; the implicit operations often can be carried out in order time rather than the naive order time typical with random dense matrices. Also, certain deterministic systems (7) have special structures that enable especially fast reconstruction in especially large problems.

In this paper, we show empirically that randomness of the matrix *A* is not required for the above phase transition phenomenon.* Namely, for a range of deterministic matrices, we show that the same phase transition phenomenon occurs with the same phase boundary as in the Gaussian random matrix case. The probability statement is now not on the matrix, which is deterministic, but instead, on the object to be recovered; namely, we assume that the positions of the nonzeros are chosen purely at random. Our conclusion aligns with theoretical work pointing in the same direction by Tropp (8), Candès and Plan (9), and especially, Donoho and Tanner (10) discussed below; however, the phenomenon that we document is both much broader and more precise and universal than what currently available theory could explain or even suggest. The deterministic matrices that we study include many associated with fast algorithms, and therefore, our results can be of real practical significance. The section *Surprises* also identifies two anomalies uncovered by the experiments.

## Methods

For each of the deterministic matrix sequences under study and each choice of coefficient set , we investigated the hypothesis that the asymptotic phase transition boundary is identical to the known boundary for Gaussian random matrices. To measure the asymptotic phase plane at a point , we chose a sequence of tuples such that and , and we performed a sequence of experiments, one for each tuple. In each experiment, we performed Monte Carlo draws of random *k*-sparse objects , attempted to recover from , and documented the ratio of successes to trials. Our raw empirical observations, thus, consist of a list of entries of the form associated with carefully chosen locations in the phase plane. This section discusses details of creation and subsequent analysis of these empirical observations.

### Deterministic Matrices Under Study.

The different matrices that we studied are listed in Table 1. A simple example is the Spikes and Sines matrix (11⇓–13), , where *F*_{n} is usual discrete Fourier transform matrix. Our list includes numerous tight frames, obeying ; several are equiangular tight frames, with the smallest possible coherence [i.e., the maximal inner product between normalized columns (12)] (14⇓–16). Not all of our frames are maximally incoherent; we consider a so-called Affine Chirp frame with coherence , which is neither tight nor equiangular. Our experiments began as assigned coursework for a course on compressed sensing at Stanford University; they considered the matrices labeled Spikes and Sines (SS), Spikes and Hadamard (SH), and Grassmannian Frame (GF), which served as simple examples of deterministic measurement matrices in that course. After our initial experiments made the basic discovery reported here, we extended our experiments to other matrices. The second stage of experiments involved the deterministic matrices labeled Paley Tight Frame (PETF), Delsarte-Goethals (DG), and Linear Chirp Frame (LC), which were extensively studied in recent years by researchers at Princeton University, especially Robert Calderbank and others. The Spikes and Noiselets (SN), Affine Plane Chirps (AC), and Cyclic matrices were added in a third stage of refinement; their selection was based on personal interests and knowledge of the authors. We emphasize that there are no unreported failures. Namely, we report results for all ensembles that we studied; we are not limiting our reports to those results that seem to support our claims, while holding back information about exceptions. The observed exceptions are disclosed in the section *Surprises* below.

### Generation of Pseudorandom Sparse Objects.

For , **R**, and **C**, a vector is called *k*-sparse if and . For case , is *k*-sparse if ; this notion was called *k*-simple in the work by Donoho and Tanner (10). We abuse language in the case by saying that entries of not in are nonzeros, whereas entries in are zeros.

In our experiments, pseudorandom *k*-sparse objects were generated as follows.

Random positions of nonzeros. The positions of the

*k*nonzeros are chosen uniformly random without replacement.Values at nonzeros. For , the nonzero entries have values i.i.d uniformly distributed in the unit amplitude set . Thus, for , the nonzeros are all equal to one; for

**R**, they are with signs chosen by i.i.d fair coin tossing, and for**C**, the nonzeros are uniformly distributed on the unit circle. For , the values not equal to zero or one are uniformly distributed in the unit interval .Values at zeros. For , the zeros have value zero. For , the zeros have values chosen uniformly at random from .

### Convex Optimization Problems.

Each Monte Carlo iteration in our experiments involves solving a convex optimization problem, in which we attempt to recover a given *k*-sparse object from . Each of the four specific constraint sets leads to a different convex program for sparse recovery of . For each specific choice of **X**, we solve

where and .

When **X** is one of , , or **R**, the corresponding can be reduced to a standard linear program and solved by a simplex method or an interior point method; in case **R**, the problem is sometimes called Basis Pursuit (17). The case **C** is a so-called second-order cone problem (18).

### Probability of Exact Recovery.

For a fixed matrix *A*, let be a random *k*-sparse object. Let , and therefore, notwithstanding the deterministic nature of *A*, is a random instance of . Define

This quantity is the same for a wide range of random *k*-sparse , regardless of details of their amplitude distributions, provided that they obey certain exchangeability and centrosymmetry properties.

### Estimating the Probability of Exact Recovery.

Our procedure follows the procedure in ref. 3. For a given matrix *A*, coefficient type *X*, and sparsity *k*, we conduct an experiment with the purpose of estimating using *M* Monte Carlo trials. In each trial, we generate a pseudorandom *k*-sparse vector as described above and compute the indirect underdetermined measurements . gives an instance of , which we supply to a solver, and obtains the result . We compare the result with . If the relative error is smaller than a numerical tolerance, we declare the recovery a success; if not, we declare it a failure. (In this paper, we used an error threshold of 0.001.) We, thus, obtain *M* binary measurements , indicating success or failure in reconstruction. The empirical success fraction is then calculated as

These raw observations are generated by our experiments.

### Asymptotic Phase Transition.

Let be an random matrix with i.i.d Gaussian entries, and consider a sequence of tuples with and . Then (**1**),

where the convergence is almost sure (10, 19⇓⇓–22).

Now let denote a sequence of deterministic matrices under study, with the same shape/sparity tuples as in the Gaussian case just mentioned. The hypothesis that we investigate is that expression **1** still holds. There is precedent: the theorem below shows, for the case , that if each is a matrix with its columns in general position in , expression **1** holds.

### Empirical Phase Transitions.

The hypothesis that marks the large *N*-phase transition boundary of each deterministic matrix sequence under study is investigated as follows. The empirical phase transition point is obtained by fitting a smooth function (e.g., a probit function) to the empirical data and finding the empirical 50% response point —the value of *ρ* solving:

Examples are in Fig. S1. Under the hypothesis that expression **1** holds not only for Gaussian matrices but also for our sequence of deterministic matrices , we have

Consequently, in data analysis, we will compare the fitted values with .

### Computing.

We used a range of convex optimizers available in the MATLAB environment:

CVX: A modeling system for disciplined convex programming by Grant and Boyd (23), Grant et al. (24), and Grant and Boyd (25) supporting two open-source interior-point solvers: SeDuMi and SDPT3.

ASP: A software package for sparse solutions by Friedlander and Saunders (26); its main solver BPdual (27) uses an active-set method to solve the dual of the regularized basis pursuit denoise (BPDN) problem based on dense QR factors of the matrix of active constraints, with only the R factor being stored and updated.

FISTA: A fast iterative soft-thresholding algorithm for solving the BPDN problem. A MATLAB implementation of the algorithm is available from the authors (5).

SPGL1: A solver by van den Berg and Friedlander (28, 29) for the large-scale BPDN problem based on sampling the so-called Pareto curve; it uses the Spectral Gradient Projection method.

Mosek: A commercial optimization toolbox that offers both interior-point and primal simplex solvers (30).

Zulfikar Ahmed also translated our code into python and used the general purpose solver package CVXOPT by Anderson and Vandeberghe (31). We verified the robustness of our results across solvers. We found that SPGL1, with the settings that we used, did not match the other solvers, giving consistently lower phase transitions; therefore, we did not use it in the results reported here. In practice, most of our results were obtained using CVX by participants in the Stanford University graduate class Stat 330/CME 362.

## Results

The data that we obtained in our experiments have been deposited (32, 33); they are contained in a text file with more than 100,000 lines, each line reporting one batch of Monte Carlo experiments at a given , , and *X*. Each line documents the coefficient field **X**, the type of matrix ensemble, the matrix size, the sparsity level, the number of Monte Carlo trials, and the observed success fraction. The file also contains metadata identifying the solver and the researcher responsible for the run. In all, more than 15 million problem solutions were obtained in this project.

Our overall database can be partitioned into several subsets, which address three general questions.

### Broad Phase Diagram Survey.

In such a survey, we systematically sample the empirical success frequency over a grid covering the full-phase diagram , including regions where we already know that we will see either all failures or all successes (an example is given in Fig. 2). For a matrix type definable only when (i.e., undersampling rate ), we considered 49 equispaced *ρ*-values in (Fig. S1 shows an example). The GF, DG, LC, and AC frames naturally allow *N* of the form for whole-number *L*; in such cases, we sampled , , etc. (Eq. **S2**). In Figs. 2–⇑4, the two-phase structure is evident.

### Precise Positioning of Phase Transition.

To address our main hypothesis regarding the agreement of phase transition boundaries, we measure at points and in the phase plane , which we expect to be maximally informative about the location of the phase transition. In fact, the informative locations in binomial response models correspond to points where the probability of response is nearly 50%; hence, we sample heavily for . Figs. S1–S3 show examples comparing the fitted phase transition with the Gaussian theoretical ones. In brief, the results are broadly consistent in all cases with the Gaussian theory, with deviations following a familiar and expected pattern.

### Finite-*N* Scaling.

The Gaussian theory is asymptotic as (19⇓–21, 34), and in general, even with Gaussian random matrices, the large *N* theory cannot be expected to match empirical data to within the usual naive SEs (21). Instead, one observes, at finite problem sizes, a finite transition zone of width and a small displacement of the empirical phase transition away from the asymptotic Gaussian phase transition of size . Analysis techniques used in ref. 21 motivated Figs. S1 and S2. Visual inspection shows that, as *N* increases, the data become increasingly consistent with the Gaussian prediction. In brief, the results are consistent with a transition zone width that tends to zero and a transition offset that also tends to zero as *N* increases.

## Discussion

### Rigorous Analysis Justifying Our Approach.

In one of four coefficient situations that we study—the case where coefficients in are real and bounded: —Donoho and Tanner (10) have proven that, for every reasonable matrix, the same probability distribution holds in finite samples. In consequence, the Gaussian theory, which describes one specific random matrix ensemble, actually describes all reasonable deterministic matrices.

### Theorem (10).

*Suppose that A is a fixed matrix with its N columns in general position*^{†}*in* . Then

This probability is independent of the matrix *A* for *A* ranging through an open dense set in the space of matrices. In addition to motivating our conclusion, it gives a valuable check on our analysis techniques, because it provides the exact expression

and the exact distribution of as binomial .

It also motivates our analysis technique. From the binomial form of , one can see that

and therefore, for any sequence of matrices all in general position, our experimental method will recover the correct phase transition:

The theorem also motivates the finite-*N* scaling analysis. Using the exact binomial law , we conclude that, for large *N*, the 97.5% success point satisfies , whereas the 2.5% success point satisfies , where denotes the *p*th percentile of the standard normal (). Hence, one sees that the transition zone between complete failure and complete success has a width roughly , thereby justifying our fitting power laws in *n* to the observed transition width.

Finally, the universality of the phase transition across all deterministic matrices with columns in general position motivates the thrust of this whole project.

### Asymptotic Analysis.

Tropp (8) and Candès and Plan (9) obtained initial theoretical results on the problem of a single large matrix. They consider a sequence of matrices and for each fixed problem size , a random -sparse , which they try to recover from measurements . Their methods apply to all of the matrix families that we have considered here, because our matrices all have low coherence. They give conditions on the aspect ratio , the coherence of , and , such that, with high probability, minimization will correctly recover . Their results are ultimately qualitative in that, for a wide variety of matrices, they predict that there will be a success phase for small . However, their results are unable to shed light on the size or shape of the success region. In contrast, we show here that the region is asymptotically the same for certain deterministic matrices as the region for random Gaussian measurement matrices. Separately, Howard et al. (35) and Calderbank et al. (36) pointed to the analogy between compressed sensing and random coding in information theory and observed that average-case reconstruction performance of a deterministic sensing matrix can be expected to be very good, even for some matrices with poor worst-case performance.

### Restricted Isometry Properties of Deterministic Matrices.

In recent work, several authors have constructed deterministic matrices obeying the so-called restricted isometry property (RIP) of Candès and Tao (2). Such matrices, if they could be built for sufficiently high *k* (relative to ) would guarantee sparse recovery in a particularly strong sense: every *k*-sparse would be recoverable. This statement would be a deterministic and not a probabilistic one. Bourgain et al. (37) are the current record holders in deterministic RIP constructions; they have constructed matrices obeying RIP for *k* slightly^{§} larger than (37). This result is still far too weak to imply anything close to the empirically observed phase transitions or the known strong neighborliness phase transition for the case of Gaussian matrices.

Calderbank et al. (36) introduced a notion of statistical RIP (StRIP) and constructed a number of deterministic matrices with StRIP. Such matrices guarantee sparse recovery in the same sense as used in this paper (i.e., according to statistical statements across random *k*-sparse objects with random positions for the nonzeros). However, to our knowledge, existing arguments are not able to derive precise phase transitions from StRIP; they only show that there is some region with high success but do not delineate precisely the regions of success and failure.

Table 1 lists several deterministic matrices obeying StRIP. Indeed, Calderbank et al. (36) used group theory to provide sufficient conditions for a sensing matrix to satisfy StRIP. Subsequently, Jafarpour (15) showed that the coherence property introduced in ref. 38 in conjunction with the tightness of the frame honors the requirements given in ref. 36 and therefore, provides sufficient conditions for StRIP. Combining these arguments, the matrices in Table 1 labeled SS, SH, SN, PETF, GF, DG, and LC obey StRIP for sufficiently large problem sizes. Refs. 15, 38, and 39 have additional discussion on the conditioning and null-space structure of these matrices. We show here that, for all these ensembles, the success region is empirically consistent with the theoretical region for random Gaussian matrices.

### Other Ensembles.

We studied several matrix ensembles not mentioned so far. For example, we used the same simulation framework and software to study random Gaussian matrices, partial Fourier matrices, and partial Hadamard matrices; our results are in line with earlier reports of Donoho and Tanner (3). We also considered several variations of the SS example based on Discrete Cosine Transforms (DCTs) of types I, II, III, and IV and the discrete Hartley transform. Finally, because of their importance to the theory of convex polytopes, we also considered the cyclic matrices when . All of the deterministic matrices that we considered yielded experimental data consistent with asymptotic agreement of the empirical phase transition and the theoretical phase transition in the Gaussian case—with exceptions noted immediately below.

### Surprises.

We were surprised by two anomalies.

#### Positive coefficients, solver for signed coefficients.

We can apply the signed solver [i.e., the solver for ] even when the coefficients are nonnegative. For several of the matrices that we considered, the phase transition that will be observed is the one that is universal for signed coefficients . However, in several cases, we observed instead a phase transition at , although the problem solved did not assume nonnegative structure. Table 2 presents a list of matrices with this positive adaptivity property. We learned from this phenomenon that, to observe universal signed behavior in the signed case, for some matrices *A*, it was necessary to make sure that the object did not always obey . For some other matrices, the signed solver gave the same phase transition regardless of whether the nonzeros contained both positive and negative values or only positive values. For conditions under which the sign pattern does not affect reconstruction, see ref. 40.

#### Cyclic matrix, positive coefficients.

Assuming *n* is even and , the cyclic matrix is defined as (41, 42) (Eq. **2**)

As discussed in ref. 42, in the case of nonnegative coefficients and assuming nonzeros in , there is a unique solution to : . Consequently, the phase transition associated with this matrix must obey for every *δ*. Empirically, we do not observe ; we, instead, observe exactly as with other matrices! (See Fig. 1.)

In short, although the theory of cyclic polytopes seemingly forbids it, our observations are consistent with large-*N* universality of the Gaussian-based formula. Remember that we are considering the behavior of numerical algorithms and that the cyclic matrix contains many very poorly conditioned subsets of *k*-columns. Possibly, numerical ill-conditioning is responsible for the failure of the predictions from polytope theory.

### Limits to Universality.

We emphasize that, although the prediction from Gaussian theory applies to many matrices, we do not expect it to apply to all matrices, the theorem quoted from ref. 10 notwithstanding.

## Conclusions

For an important collection of large deterministic matrices, the behavior of convex optimization in recovering random *k*-sparse objects is accurately predicted by the theoretical expressions that are known for the case of Gaussian random matrices. Evidence is presented for objects with coefficients over each of the sets when the convex optimization problem is appropriately matched and the positions and signs of the nonzeros are randomly assigned.

Standard presentations of compressed sensing based on RIP suggest to practitioners that good deterministic matrices for compressed sensing are as yet unavailable, and possibly will be available only after much additional research. We use instead the notion of phase transition, which measures, in a straightforward way, the probability of exact reconstruction. We show here that reconstruction of sparse objects by convex optimization works well for certain deterministic measurement matrices—in fact, just as well as for true random matrices. In particular, our demonstration covers several explicit deterministic matrices for which fast transforms are known.

## Acknowledgments

We would like to thank Michael Saunders and Michael Friedlander for providing us with a prerelease version of the optimization software ASP. This work was partially supported by National Science Foundation Grant DMS 0906812 (American Reinvestment and Recovery Act). Part of the work was done when S.J. was a research scientist at Yahoo! Research. Sam Gross held an NSF-DMS VIGRE fellowship, and Young Choi was supported by Army Grant W911NF-07-2-0027-1. Kevin S. Raines was supported by National Science Foundation Grant DGE 1147470 as well as the Stanford Humanities and Sciences Fellowship. Mainak Choudhury was supported by a 3Com Corp. Stanford Graduate Fellowship. Carlos Sing-Long was partially supported by a Fulbright-CONICYT scholarship.

## Footnotes

↵

^{1}A complete list of the Stat 330/CME 362 Collaboration can be found in*SI Text*.- ↵
^{2}To whom correspondence should be addressed. E-mail: donoho{at}stat.stanford.edu.

Author contributions: D.L.D. designed research; H.M., S.J., M.G., S.3.C.3.C., and D.L.D. performed research; S.J. and D.L.D. contributed new reagents/analytic tools; H.M. and D.L.D. analyzed data; and H.M., M.G., and D.L.D. wrote the paper.

The authors declare no conflict of interest.

Data deposition: The data reported in this paper are available at http://purl.stanford.edu/wp335yr5649.

See Commentary on page 1146.

↵*This research began as a class project at Stanford University by students of Stat 330/CME 362, which was taught by D.L.D. in Fall of 2011 (Teaching Assistant: M.G.). The basic discovery was a joint effort of all of the participants. Independently, S.J. made a similar discovery in studying Delsarte-Goethals frames for his thesis.

↵

^{†}A collection of vectors in is said to be in general position if no subcollection of, at most,*n*vectors is linearly dependent.↵

^{§}I.e., , with .This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1219540110/-/DCSupplemental.

Freely available online through the PNAS open access option.

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