We know how to represent the solution of a nonogram as a sparse vector . Let us now design the constraints; they will be linear equations and linear inequalities that the entries of must satisfy according to the nonogram. The only information that the nonogram provides are the row and column sequences, so we want to make the most of it. Afterwards, we obtain the sparse vector by minimizing its 1-norm under these constraints. I will discuss the code and the results in the next and final post.
3. Introducing compressive sensing
Today, we will consider how to turn the solution of a nonogram into a sparse vector. But first, let me take a detour and illustrate an approach that Oscar and I initially had considered and comment on its strengths and weaknesses in solving the puzzle.
Throughout our discussion, we will consider the stylish lambda example:
Over the next month, I want to present a project that my friend Oscar and I came up with and worked on for a class. We developed a new technique to solve nonograms using compressive sensing.
The trick is knowing how to write the solution to any nonogram as a sparse vector . Sparse means there are many entries that are zero, and compressive sensing considers this fact to find a unique solution to an underdetermined system! It’s strikingly different from traditional linear algebra, which says that an underdetermined system that has a solution has, in fact, infinitely many solutions.
Our method avoids (1) partial fill-ins, (2) heuristics, and (3) over-complication, and only requires that we solve a binary integer programming problem. There was one issue that Oscar and I couldn’t solve in time, however. I hope that I can now by revisiting this project after three years. Ideas are welcome!
This is a right whale. Right now, there are only 450 right whales alive in the North Atlantic Ocean. They are the rarest among all large whales.
But fear not. A month ago, a group of people came to their rescue, and those people? They were data scientists.
Our final problem has no known exact solution. We want to find the area of the shape formed by,
The inequality has two parameters and . They are quantities of length, so they take on a nonnegative value. Let’s try out some values and see what these parameters do. I have colored the resulting shapes in red. (Note, the scales are different.)
When (move along the diagonal, starting from bottom-left), we get the shape of a nice heart. When we increase while holding fixed, the heart morphs into a circle. On the other hand, when we increase while holding fixed, the heart turns into two petals. Well, I see bunny ears.
Clearly, the shape (i.e. area) of our heart depends on the radius and the ear length . Can you guess the formula for the area ?