Adaptive Algorithm
from Keio's Faculty of Science and Technology Pursuit of an ultimate filter
that extracts desired information from signals

What is an adaptive filter that extracts and elucidates useful information from signals?

Our society is brimming with truly diverse signals (information), such as visual images, sounds, radio waves, brain waves and stock price fluctuations, to name a few. The primary role of signal processing is to convert these signals into numerical form and extract only a desired piece of information for identifying the characteristics of a given phenomenon or using it to make a projection. By taking a mathematical approach and making complex signals obtained from diverse phenomena in the universe easily usable, Associate Professor Masahiro Yukawa is intent on establishing a mathematical system useful for extracting needed information.

“Suppose our modern world loses signal processing, we will suddenly find our life totally inconvenient and boring – no TVs, no mobile phones, no digital cameras and no airplanes flying in the sky. MRI, the medical equipment, is also a product of signal processing. Popular topics of our conversation of late, such as drones, androids and big data analysis, are also impossible without signal processing. In short, signal processing is something essential to our modern life just like air,” mentions Dr. Yukawa.
For example, the microphone converts cellular phone sound into and the image sensor converts images captured by a digital camera into electrical signals, respectively. By replacing these electrical signals with numeric sequences, they come to be mathematically processed and become objects of the so-called signal processing.
“What I have actually focused my effort on is adaptive processing of acoustic and radio communication signals. One example is the echo canceler. In videoconference or cell-phone conversations, your voice may echo back to yourself belatedly via the other side’s microphone, which is often annoying. The echo canceler is designed to cut the echoed voice only. Because your echoed voice is not necessary, signal processing performs subtraction and allows only the other side’s voice to be delivered. Likewise, in the case of radio communication, adaptive signal processing deals with multiple communication data, which simultaneously arrive in the base station, separate and deliver them to individual users. What is indispensable to such signal processing is the “adaptive filter.”

Just like a coffee filter, the adaptive filter extracts, through a mathematical operation called “filtering”, an exactly needed piece of information.
“Speaking of sounds, the role of the adaptive filter is to estimate the function f capable of extracting the required sound x from the observed sounds y. Putting it another way, we simply have to seek the function f because we need to estimate x from y,” explains Dr. Yukawa.
As the term “adaptive” suggests, however, we, operators, must work – flexibly adapting the filter function to the constantly changing environments. In other words, the adaptive filter’s performance depends largely on our ability to express, in real time, a function made as closer as p ossible to the required information while minimizing computational complexity.
Here’s a major problem: most of actual phenomena are of nonlinearity, which is difficult in general to be handled mathematically. Nonlinearity means not being linear. In short, with these phenomena, input and output are not proportional to each other. Attempting to deal with nonlinear data causes the amount of computation to suddenly increase, making it difficult to extract the needed information speedily.
“Volterra filter and neural network are typical approaches to handling nonlinear data. But the former involves enormous volumes of computation while the latter has the drawback of ending up with a local minimum. To address these problems, I decided to adopt what is known as the “kernel adaptive filter” based on the kernel method.
The kernel method allows data to be mapped onto a feature space of higher dimensions (for example, from two dimensions to three dimensions), thereby converting a group of data into an easierto-handle group. This technique is widely used for facial recognition and other pattern recognition as well as big data analysis. Formerly it was used in batch processing, a method to process batches of collected data all at once. In recent times, it is increasingly in use for online scenarios in which new data are collected and processed from moment to moment.
“ I would omitexplanation of reproducing kernel because it is too technical and difficult. But the benefit of reproducing kernel exists in that it can evaluate a function value as an “inner product.” A simple example of an inner product is the sum of all products that have been obtained by multiplying two vector elements in series. The inner product makes it possible to express correlations between two vectors. Once these correlations can be expressed with an inner product, we can now use our knowledge of linear models to the fullest,” Dr. Yukawa states emphatically.
In short, you can say that the greatest advantage of this method is that it allows us to use the easy-to-compute linear theory in dealing with usually difficultto-handle nonlinear data.

Dr. Yukawa has developed the “multikernel adaptive filter,” based on the kernel adaptive filter and making the most of the latest developments in convex optimization.
“To put it in a general image, the filter I developed approximates a function’s waveform by adding up bell-shaped curves known as Gaussian kernels. We prepare in advance a number of Gaussian kernels – wide and low curves, narrow and high curves, etc. – then arrange coefficients, which determine their heights, in matrices. Finally, we estimate the function by adding up a smaller number of peaks.”
“The key point here is to include a mechanism designed to automatically identify peaks that fit the function you wish to estimate while nullifying coefficients for most of the other peaks. Sparse matrices refer to those matrices that contain many zeros. Neat arrangement of information (in sparse matrices) enables optimal model selection.”
This allowed us to adjust Gaussian kernel widths to nonlinear function forms adaptively and automatically and respond freely to function forms even when they may change with a lapse of time. Highly accurate estimate is thus possible with a smaller number of peaks.
“By using this technique, for example, we can make highly accurate, real-time predictions of future power outputs in solar power generation on the basis of power output data of the past,” he adds. The “fixed-point theory for nonexpansive mapping” is the mathematical foundation for Dr. Yukawa’s study.
“A fixed point refers to a point that does not move even when mapping T is operated, that is, x such that Tx = x. Also, nonexpansivity refers to the characteristic which the operation of a mapping does not expand the distance between two points. Today, it is becoming increasingly known that solutions to problems – in various fields of science and engineering – can be expressed as fixed points of a mapping. Actually, the use of a nonexpansive mapping makes it easy to design fixed points, or algorithms to seek the solution of a problem. The multikernel adaptive filter estimates unknown functions by expressing everchanging functions as fixed points of a mapping which is created using data that flow in from moment to moment,” he remarks.
Dr. Yukawa expresses his desire to create an “ultimate filter” in the future – the ultimate filter capable of dealing with any complex phenomena. After 2015 set in, Dr. Yukawa is making a name for himself internationally as three of his papers were already accepted for publication in the journals of IEEE (The Institute of Electrical and Electronics Engineers, Inc.) and he was picked out as an Associate Editor for the world premium journal in the field of signal processing. He continues to focus on fundamental mathematical systems underlying the currently highlighted research themes such as distributed signal processing, big data analysis and deep learning while taking a side glance at them.

As far as I was told by my mother, I was a very talkative boy, who often reported to my nursery school teacher about everything that had happened at home the day before. To make the matter worse, I spoke out clearly and in an easy-to-understand way. Later my mother complained to me that she had felt pretty embarrassed (Laughter).
I have a sister four years older than me. As a small boy, I used to follow her and play house with her. This gave me a nickname “shadow-like follower” (Laughter).
I was born in Minami-Ashigara City, Kanagawa Prefecture. Both my father and mother were public servants, father working for the Odawara City Office and mother for the Ministry of Finance’s Printing Bureau. Both of my parents were good at math in their school days. Presumably having inherited their genes, they say I was good at mental calculation since my nursery school days. But I have few memories of those days. The only thing I still member is that, as an elementary school boy, I often went shopping with my mother, when I did sums in my head.
Because of my mathematical interest, I was very fond of riddles, quizzes and puzzles and loved reading books about mathematics.
Soon after entering the elementary school, I attended a soroban school. Thanks to a good teacher, I liked the art of soroban and continued to learn it up to the end of junior high school second year. I hold a first dan license in soroban and a third dan license in mental calculation. On the occasions of soroban competitions, I was always among the best; at an All Kanagawa Prefectural soroban competition when I was an elementary school sixth grader, I became the champion in the category of calculating figures read off aloud. Even today I still do two-digit addition and multiplication by mental calculation – not so well as in the past, though.
I used to teach arithmetic and mathematics to my friends. Looking back now, I may also have learned something by teaching them. I’m still grateful to my friends.

I don’t think so. An expanse of rice fields could be found around my house, so rice and flower fields were ideal playgrounds for us children. Actually I was an active elementary school boy; the moment recess time began, I dashed out of the classroom into the schoolyard to enjoy ball playing with classmates. There was also a period when I went to school at 6:00 in the morning just to practice long-distance running together with my friends – a cherished memory now.In my junior high school days, I served as captain of the table tennis club and even participated in the prefectural tournament. The more I exercised, the better I slept at night (Laughter).
Meanwhile, at one time I was crazy about “Famicon” video game machine and “Game Boy” handheld game device. I played and played these games until I got sick of them. Having done too much, I said good-bye to these games (Laughter).

No, they didn’t. My parents never criticized me about my attitude about studies because I did at least what I should do – I did homework as assigned and listened carefully to classroom lessons. I’m grateful to my parents for basically allowing me to do whatever I liked. When I was an elementary schoolboy, I was making a plastic model and badly soiled the carpet surface here and there with a sticky bond. Even at that time I remember they didn’t scold me at all. As parents they must have been very generous.
Following entry to a local junior high school, I began to attend a small cram school in my neighborhood partly because of my elder sister’s influence. Thanks to the cram school teacher’s enthusiastic guidance, I became more and more interested in mathematics. In those days, however, I wanted to become a certified public accountant in the future, not a researcher.
As for senior high school, I chose Prefectural Atsugi High School, which was outside of my school district. It was my routine to make a 40-minute trip one way on the Odakyu Line train while reading the mathematics textbook on board. Presumably, this experience may have made my backbone as a researcher as I am today.

I did so partly because of recommendation by a father of my friend and partly because of advice by a tutor at my prep school. By this time, I became inclined, vaguely though, to choose in the future a career based on IT-related or mathematical knowledge.
To be honest, however, I was not so serious about study while I was in the lower undergraduate grades. Although I knew I was interested in theory-oriented subjects, such as on Fourier transform and Laplace transform, I was still unable to narrow down my academic interest. Under such circumstances, as an undergraduate I focused on club activities and a part-time job at a fast food restaurant. The fast food restaurant provided me with an opportunity to interact with students from other universities; for me working there was as enjoyable as a club activity at university.
Another pursuit I took up after entering the university was dancesport. While inspired by a TV program as the first opportunity, I can say that something minor, or something that not many people are doing, caught my interest. The dancesport club had only four or so male members for each undergraduate grade – really minor wasn’t it (Laughter)? In this sense, it seems to have something in common with soroban and table tennis. Turning to my research side, signal processing is quite popular in the world, but I’m focusing on themes on which few people have an eye on.
My specialty in dancesport was Latin dance. I continued this pursuit for about ten years. Some jokingly say that I chose England as my study-abroad destination because it is the home of dancesport. The fact is that in England I participated every week in a more elegant ballroom dancing practice session, not dancesport … Meanwhile, I went to watch dancing competitions from time to time and I was particularly get excited when I visited Blackpool, which is the place of the Blackpool Dance Festival, to watch an amateur competition. At the time, the TV show “Strictly Come Dancing” was winning great popularity among British viewers.

In the beginning yes, but very soon I got accustomed to dancing with a female partner. To tell the truth, soon after marriage I invited my wife to enjoy ballroom dancing with me, but she gave up after only several times. Her excuse was that she got fed up with the way I instructed, which was too strict and specific … (Laughter). I didn’t mean to be too strict with her, but it seems that the habits I had acquired through dancesport lingered, which urged me to specifically advise her on posture, shift in the center of gravity and so on. In everything, once I get started, I tend to forget myself, which, I admit, is my forte as well as my shortcoming.
It was when I became a senior and joined a lab specializing in signal processing and communication theory that I became hooked on the attraction of research work. The lab I initially had in mind was one focusing on computer vision, but the number of applicants was more than the capacity the lab could accommodate. Playing “rock paper scissors" was the only way to decide new members, so I gave up the game and switched my wish. Looking back now, this change to the theory-oriented lab proved to be the right answer.

Ever since my childhood, I like eating; so each meal is the best breather for me. Next, my blissful time is when I listen to jazz or Latin music while enjoying sweets together with a cup of coffee that I’ve brewed for myself. My routine on the campus is to go to the cafeteria together with my students. I also enjoy a coffee break with them, which is enjoyable.

Keio has a well-developed public relations system effectively encouraging us younger researchers in research activities, for which I’m truly grateful. At Keio, even lower-grade students have easy access to professors – or an easy distance between students and professors, you might say. This is an enviable environment. Keio also offers a variety of student support systems, allowing them to become conscious of their future courses at an early stage. This is another strong point of Keio.

Student : Dr. Yukawa has a cool, objective eye to research, yet demonstrates extreme enthusiasm when striving to achieve research goals. His pet phrase is: “Let’s make our lab No. 1 in the world!” Our lab is a pleasant place to be in, always brimming with conversations and laughter – exchanging ideas and developing discussions during lunch meetings or coffee breaks.