A Model is Worth Tens of Thousands of Examples

Abstract

Traditional signal processing methods relying on mathematical data generation models have been cast aside in favour of deep neural networks, which require vast amounts of data. Since the theoretical sample complexity is nearly impossible to evaluate, these amounts of examples are usually estimated with crude rules of thumb. However, these rules only suggest when the networks should work, but do not relate to the traditional methods. In particular, an interesting question is: how much data is required for neural networks to be on par or outperform, if possible, the traditional model-based methods? In this work, we empirically investigate this question in two simple examples, where the data is generated according to precisely defined mathematical models, and where well-understood optimal or state-of-the-art mathematical data-agnostic solutions are known. A first problem is deconvolving one-dimensional Gaussian signals and a second one is estimating a circle’s radius and location in random grayscale images of disks. By training various networks, either naive custom designed or well-established ones, with various amounts of training data, we find that networks require tens of thousands of examples in comparison to the traditional methods, whether the networks are trained from scratch or even with transfer-learning or finetuning.

A Pointflow Implementation Details

The pointflow dynamics are implemented by discretising time and approximating the time derivative with a forward finite difference scheme, although it could be improved with a Runge-Kutta 4 implementation [5]. Given the small magnitudes of the fields, we found that a large time step $dt = 50$ works well. We define three thresholds, $\tau _l = 0.9$ for $C_l$, $\tau _s = 10^{-6}$ for $C_s$, and $\tau _{len}=0.001$. we consider having looped $C_l$ if a point reaches a previous point within squared Euclidean distance $\tau _l$ while having on the trajectory between the looping points at least one point with squared distance to them of at least $\tau _l$. A trajectory is stuck if it reaches a point where the current flow V has small magnitude $\Vert V\Vert _2^2\le \tau _s$. Each flow is run for $N_i = 1000$ iterations, and trajectories shorter than $\tau _{len}$ are discarded, e.g. trajectories of type $C_s$. We used $\sigma _{Pf} = 5$ for blurring out the noise before computing the fields. The implemented pointflow algorithm for finding contours in our circle images is presented in Algorithm 1.

After computing the list of contours $\mathcal {C}$ in the image I, we estimate the radius using the average curve length $\hat{r} = \tfrac{1}{2\pi }\sum _{i=1}^{|\mathcal {C}|} \textrm{length}(\mathcal {C}_i)$. Since the average of the points did not yield the best estimation of the circle centre, we estimate it instead using least squares. The equation of a circle is naturally given by $(x-c_x)^2 + (y-c_y)^2 = r^2$, which can be written as $\theta _1x + \theta _2y + \theta _3 = x^2 + y^2$, where $\theta _1 = 2c_x$, $\theta _2 = 2c_y$, and $\theta _3 = r^2 - c_x^2 - c_y^2$. We can thus estimate for each contour $\theta = (\theta _1, \theta _2, \theta _3)^\top$ by least squares as $\hat{\theta } = A^\top (AA^\top )^{-1}B$, with $A_{i,:} = (x_i, y_i, 1)$ and $B_i = x_i^2 + y_i^2$ and i ranging in the number of computed points on the contour. From $\hat{\theta }$ we can estimate $\hat{c} = (\tfrac{\theta _1}{2}, \tfrac{\theta _2}{2})$. The final centre estimation is then given by the average of this estimation over all contours. Note that we can also estimate r using $\theta _3$ but we found that it did not outperform the lenght strategy so we do not use it.