From bc48ba1033d83c79e23a95e99cb17b2bd0e4b500 Mon Sep 17 00:00:00 2001 From: Erik Strand <erik.strand@cba.mit.edu> Date: Wed, 8 May 2019 22:44:53 -0400 Subject: [PATCH] Answer 15.7 --- _psets/12.md | 75 ++++++++++++++++++++++++++++++++++ assets/img/pset12_fig_a.png | Bin 0 -> 44427 bytes assets/img/pset12_fig_b.png | Bin 0 -> 22295 bytes assets/img/pset12_fig_c.png | Bin 0 -> 45007 bytes assets/img/pset12_fig_d.png | Bin 0 -> 42786 bytes assets/img/pset12_fig_e.png | Bin 0 -> 51321 bytes assets/img/pset12_fig_e_2.png | Bin 0 -> 67061 bytes assets/img/pset12_fig_f.png | Bin 0 -> 46895 bytes assets/img/pset12_fig_f_2.png | Bin 0 -> 43671 bytes assets/img/pset12_fig_g.png | Bin 0 -> 20629 bytes assets/img/pset12_fig_g_2.png | Bin 0 -> 48445 bytes assets/img/pset12_fig_g_3.png | Bin 0 -> 19388 bytes assets/img/pset12_fig_g_4.png | Bin 0 -> 46708 bytes 13 files changed, 75 insertions(+) create mode 100644 assets/img/pset12_fig_a.png create mode 100644 assets/img/pset12_fig_b.png create mode 100644 assets/img/pset12_fig_c.png create mode 100644 assets/img/pset12_fig_d.png create mode 100644 assets/img/pset12_fig_e.png create mode 100644 assets/img/pset12_fig_e_2.png create mode 100644 assets/img/pset12_fig_f.png create mode 100644 assets/img/pset12_fig_f_2.png create mode 100644 assets/img/pset12_fig_g.png create mode 100644 assets/img/pset12_fig_g_2.png create mode 100644 assets/img/pset12_fig_g_3.png create mode 100644 assets/img/pset12_fig_g_4.png diff --git a/_psets/12.md b/_psets/12.md index ee34eaa..83aeb66 100644 --- a/_psets/12.md +++ b/_psets/12.md @@ -222,12 +222,20 @@ the convolutional encoder in Figure 15.20, what data were transmitted? {:.question} This problem is harder than the others. +My code for all sections of this problem is +[here](https://gitlab.cba.mit.edu/erik/compressed_sensing). I wrote it in C++ using +[Eigen](http://eigen.tuxfamily.org) for vector and matrix operations. + ### (a) {:.question} Generate and plot a periodically sampled time series {$$t_j$$} of N points for the sum of two sine waves at 697 and 1209 Hz, which is the DTMF tone for the number 1 key. +Here's a plot of 250 samples taken over one tenth of a second. + + + ### (b) {:.question} @@ -244,18 +252,28 @@ D_{ij} = \end{align*} $$ + + ### (c) {:.question} Plot the inverse transform of the {$$f_i$$} by multiplying them by the inverse of the DCT matrix (which is equal to its transpose) and verify that it matches the time series. +The original samples are recovered. + + + ### (d) {:.question} Randomly sample and plot a subset of M points {$$t^\prime_k$$} of the {$$t_j$$}; you’ll later investigate the dependence on the sample size. +Here I've selected 100 samples from the original 250. The plot is recognizable but very distorted. + + + ### (e) {:.question} @@ -270,6 +288,19 @@ $$ {:.question} and plot the resulting estimated coefficients. +Gradient descent very quickly drives the loss function to zero. However it's not reconstructing the +true DCT coefficients. + + + +To make sure I don't have a bug in my code, I plotted the samples we get by performing the inverse +DCT on the estimated coefficients. + + + +Sure enough all samples in the subset are matched exactly. But the others are way off the mark. +We've added a lot of high frequency content, and are obviously overfitting. + ### (f) {:.question} @@ -287,6 +318,20 @@ $$ {:.question} and plot the resulting estimated coefficients. +With L2 regularization, we remove some of the high frequency content. This makes the real peaks a +little more prominent. + + + +However it comes at a cost: gradient descent no longer drive the loss to zero. As such the loss +itself isn't a good termination condition. In its place I terminate when the squared norm of the +gradient is less than $$\num{1e-6}$$. The final loss for the coefficients in the plot above is +around 50. + +You can easily see that the loss is nonzero from the reconstructed samples. + + + ### (g) {:.question} @@ -301,3 +346,33 @@ $$ {:.question} Plot the resulting estimated coefficients, compare to the L2 norm estimate, and compare the dependence of the results on M to the Nyquist sampling limit of twice the highest frequency. + +With L1 regularization the DCT coefficients are recovered pretty well. There is no added high +frequency noise. + + + +It still can't drive the loss to zero. Additionally it's hard to drive the squared norm of the +gradient to zero, since the gradient of the absolute values shows up as 1 or -1. (Though to help +prevent oscillation I actually drop this contribution if the absolute value of the coefficient in +question is less than $$\num{1e-3}$$.) So here I terminate when the relative change in the loss +falls below $$\num{1e-9}$$. + +The final loss is around 40; smaller than we were able to find with L2 regularization. However it +did take more effort: this version converged after 21,060 iterations, as opposed to 44 (for L2) or +42 (for unregularized). + +The recovered samples are also much more recognizable. The amplitude of our waveform seems overall +a bit diminished, but unlike our previous attempts it looks similar to the original. + + + +This technique can recover the signal substantially below the Nyquist limit. The highest frequency +signal is 1209 Hz, so with traditional techniques we'd have to sample at 2418 Hz or faster to avoid +artifacts. Since I'm only plotting over one hundreth of a second, I thus need at least 242 samples. +So my original 250 is (not coicidentally) near here. But even with a subset of only 50 samples, the +L1 regularized gradient descent does an admirable job at recovering the DCT coefficients and +samples: + + + diff --git a/assets/img/pset12_fig_a.png b/assets/img/pset12_fig_a.png new file mode 100644 index 0000000000000000000000000000000000000000..9a732c8a7b3f6de4fbab9a098287dcb545387981 GIT binary patch literal 44427 zcmeAS@N?(olHy`uVBq!ia0y~yU}|7sV0^&A#=yW}dhyN^1_lO}VkgfK4h{~E8jh3> z1_lPs0*}aI1_u5_5N2FqzdVzHfx)uGHKHUqKdq!Zu_%=xH?gE3C%+^oGfAN=wWv5V zKTp9}&p^*uM<Ju6q`*pFAE7`mzbIXQF8Aui3=9eko-U3d6?5Lqt(+s1I`_Z*)w{;F zmyItM%kO;O!l2lwU>K-S_&7;GAna*C&w@vck%^BFo!`MF_|4U2y0_SLZ_aj)mZLAY zSezyluuhRtVJS+V<Z1N%>gBt}wb$PFzgoV^`m*ii(AcZTKiAw}e|=BFuUWIa&IZqa z{&}xT<g`=)7Do^|lyLehBZ$G`C~&itMFA|>k}!cmg`-J<qe*~Ar~#s|$&qP-0!Yls zz=J^uEXKjc=>Srsz;RH4(F1IVqJ#<qCs>T7c_bQks3dHC9Ixjjp1tL7Z*85R!g==W z*=KSgGH1`8ouI;*eSO{5H|bB9C#(A(Grb-&`SoE14kdm?Mh}6OQ}6EXF8q2mJo(O! z!UC(g64qrp`~H1hpZxvZ-JNz@=3KmUrzd;;Ua`x|e4RZ#IcLq9RdBEJ`NM6u^G<(V 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