"A house is not a home unless it contains food and fire for the mind as well as the body", Benjamin Franklin once said. This sentence might be true for those who don't know what people with special needs are looking for. When you live with a person who faces mental-health issues, all the books on the shelves won't keep him/her warm. Special needs housing looks more like this quote from Ralph Waldo Emerson: "A house is made with walls and beams; a home is built with love and dreams." The feeling they get inside their homes is far more important than how it looks.
Who should get more involved and care more about the poor and disabled if not the biggest corporations who feed on the work of the healthy and many? We are yet to see the human side of the banking system. We are yet to see more social involvement from these arrogant giants. Until now, they have turned a blind eye. Have you heard of banks investing their easy profits, their ASSETS, in special needs housing? No. I would more happily pay my mortgage ON TIME every month knowing that part of that interest provided a warm meal to someone, a shelter for the homeless, and education for someone with special needs instead of paying for ATM service bills, overpriced rents, and wasted utilities. After so many years of loans and mortgages, how is it possible to have over 510,000 homeless people in the US? That means that they could have a town of their own in every state – 10,000 people each if they decided to group.
But no, banks take home all the earnings. They are the pulse of the economy and alter the S&P 500 index, so they can't mess it up by getting involved in "small" social projects. "Earn more and care less" is their guiding adage.
In the meantime, down here, in the real world, families with special needs members who want to own a house must go the traditional way, with a 25- or 30-year mortgage. Or rent. But banks, as the largest providers of mortgages, get involved neither in the lives of young people living with a disability nor in the lives of older disabled people. Where is the love? Where are the dreams? …the dreams with a deadline?
She was there again. Eyes wide open in front of the narrow, rugged door of the old elevator, hands clenched on the push handles of her daughter's new wheelchair. She smiles today because the lift is not out of service. Otherwise… Every time the doors open Jane recalls that painful day when she struggled to bring Ania into the world. After a long labor, the doctors gave her the paralyzing news: Ania had suffered hypoxia, and her brain might be damaged. To what extent, Jane had soon to find out. But their lives have never been the same again. Year after year, Jane and her husband learn to live with a daughter with special needs, but they have yet to find a better place to live. Jane can stand that elevator door no more. She will soon meet with a real estate agent specialized in housing for families with disabled children like her.
We hear many stories like this, and we care about them all, that's why we try to find the best residential communities for adults with special needs. Before we look at what is special needs housing and what it should entail, let's see what "special needs" mean.
UN Convention on the Rights of Persons with Disabilities (UNCRPD) recognizes that disability is an evolving concept. When you say that someone has special needs, in most cases it means that they have some sort of difficulty which can be overcome only with the help of others – a family member, a caregiver, or a medical assistant. People with special needs have specific health problems that don't allow them to live completely independent or unsupervised. But to be more exact, we should borrow the definition of disability from the American Community Survey, which looks at six crucial health aspects: hearing disability, visual disability, cognitive disability, ambulatory disability, self-care disability, and independent living disability. And of course, there is the PTSD disability – the post-traumatic stress disorder, with almost 1 million cases. Ania, for example, couldn't walk and had cerebral palsy, so she needed someone to help her move around and enjoy the little things most of us take for granted.
Disability is part of our human experience. Not everyone with a special need has been born that way. We must keep in mind, though, that everyone might experience it later in life, either permanently or temporarily.
How are people with disabilities doing financially?
I could hear you utter "Poor Jane!" as you were reading her story… And you know what? You were right! She is poor! According to Cornell Disability Statistics, Ania is among the 26.1 % non-institutionalized persons with a disability who were living below the poverty line in 2017. A large majority of people with disabilities are either unemployed, under-employed or earn lower wages. The actual numbers don't even matter because we wouldn't be taken by surprise. We, unfortunately, see a link between disability and poverty, and for sure, there is a strong one.
To add insult to injury, though, let me tell you that individuals with disabilities are unable to save money for an emergency account, so 70% of individuals with disabilities responded that they might not be able to come up with $2,000 to meet an unexpected expense. The situation gets worse among the elderly with disabilities. These aside, there is this fear of losing SSI or SSDI financial support. To qualify for these aids, people with disabilities must not have more than $2,000 in their bank accounts, or $3,000 as a couple. According to the Social Security Administration, SSDI stands for Social Security Disability Insurance and is paid to those who are insured. Supplemental Security Income, or SSI, is paid based on financial need.
So, if our administration is regulating special needs funding so tight, the same administration can adopt laws that make everybody rich overnight. But the struggle with poverty is too profitable for the banking system. As long as everybody is in the red, there will be room for even more debt – for many, a sad truth; for the few "chosen", a profitable business.
Studying personal finances is a way of fighting for financial freedom. There are many books on this topic, and a very well known author is Dave Ramsey, for sure. But when it comes to helping adults with special needs and their caregivers get better control over their monthly income and expenses, National Disability Institute got really involved and prepared an Inclusive Financial Education Toolkit which is available for free to download. I know it's hard to accept, but if you really want to take saving money seriously, then you can make it work around a strict budget by learning to focus on needs rather than wants.
Access to education is important for children with special needs and their parents as well. For example, a parent whose child was diagnosed with autism will want to learn more about this condition and how to accept it and live with it, while taking their child to occupational therapy, speech therapy or physical therapy.
There are kids with learning disabilities (LD) as well – one in three parents is not prepared to take on the challenge, and only a third are optimistic about their ability to cope. Learning disability puts kids at higher risks for bullying. 66% of kids with learning disabilities are boys. Among the common learning impairments are dyslexia, dysgraphia, dyscalculia, and dyspraxia. Almost half of the adults with LD are employed and usually earn $25,000 or less per year.
Besides learning disabilities, which prevails among students ages 6-21, the next are speech impairment, intellectual disabilities, autism, emotional disturbance, developmental delay, hearing impairments, orthopedic impairments, visual impairments, traumatic brain injury and deaf/blindness. Mental disorder is a hardship for 1 in 5 kids.
According to the Individuals with Disabilities Education Act (IDEA), enacted in 1975, all children with disabilities between 3-21 years must be provided free public school education, depending on their needs. However, 3.1% of infants and toddlers birth through age 2 received early intervention services in 2016. Another 6 million students (9%) ages 6 through 21 have received education under IDEA, Part B. More than half of the students (63.1%) have received education inside a regular class, and less than 3% attended a separate special needs school. Over 90% of the students participated in a state math assessment. The good news is that 67.4% of students ages 14 through 21 who complete education under IDEA, Part B, graduate with a regular high school diploma. However, the percentage of students who drop out of this educational scheme is 17.5%. But many exist special education for good reasons, like to attend regular school (almost 10%).
So, the situation is not too dramatic. Most students with a disability turn out to be employable and could achieve a certain level of independence. And as we will see in the next section, more and more companies are aware of this reality and are opening positions for people with disabilities.
Do people with disabilities work?
Yes! Most people with a moderate disability can find a job, and 44% of them are employed full time, year-round. More and more companies see people with disabilities as being not inferior but unique in the skills they have. Companies that hire people with various impairments are Home Depot, Target, McDonald's, Walmart, Goodwill, Ikea, Stop and Shop, Acme and many more.
When independence is not possible, the perspective is dull. It is almost impossible for a person with a severe disability to get hired. Statistics published by the US Bureau of Labor pointed out that 18.7 percent of persons with a disability worked in 2017. Most men and women with disabilities were employed in nonagricultural industries like education and health services, retail trade, and manufacturing. The Government employs 14.4% of those who live with a less severe disability while only 10% choose to take on greater responsibility and work as self-employed.
Their productivity is usually lower and might work on minimum wage. Even so, they risk losing some kind of financial assistance due to "higher incomes." Then, discrimination is another issue. Women with disabilities face double discrimination. But these are problems faced by people who have no disability as well. A lot needs to be done in order to integrate people with disabilities into their communities fully. But it would be useless to try to discuss here why most parents of children with disabilities think it would be better for their children to live and work together, in one place. Everybody has a different opinion on how the life of people with various impairments should look like. One thing is for sure: they all need a place to live, so let's see how they are doing in this area.
The U.S. Census Bureau, American Housing Survey reveals that there are currently 17.2 million households with at least one disabled person, 22.2% of the total number of households. But in 2015, more than 40 million people were living with a disability in the United States. These numbers mean that in some homes there are more people with special needs.
In 2017, there were 1.7 million people with disabilities living in a manufactured or mobile home and 506,000 households with a disabled person lived in 1-bedroom units. But most families who care for a special needs individual live in 3-bedroom units – 8.85 million households, actually.
If we change the perspective, we find that another 9.8 million households with a disabled person are renters, and almost half of them have an annual income of less than $20,000. According to the Fair Housing Act and the Fair Housing Amendments Act (42 U.S. Code §§ 3601-3619, 3631) landlords are not allowed to refuse to rent their property solely by reason of renter's disability. They must do everything to accommodate the needs of a disabled child or adult, giving them equal opportunity to use and enjoy their home as well as the common spaces.
But the government came with solutions for those who rent as well. What a relief for so many families to know they might be eligible for public housing programs, rental assistance or subsidized housing, and Housing Choice (Section 8) voucher programs. Others may qualify for a Non-Elderly Disabled (NED) Voucher. You can find more information regarding housing for special needs adults at your local Housing Agency.
So, the government understands the correlation between poverty and disability and tries to fill in the gaps, but it is limited by other's economic interests. Then the NGOs (Non-Governmental Organizations) try to cover what the central administration can't. There are quite a few non-governmental organizations dedicated to improving the lives of people with disabilities and most of them are listed on Disabled World.
In 2016, of the disabled population, 41.4% were 65 and older. As the population is getting older and living longer, the number of people getting a disability also increases. Three times more people after the age of 65 find themselves among the people with disabilities. And their houses are not prepared for that. Moreover, by 2060, there will be almost 100 million people 65 and older living in the US, according to Senior Living.
Have you ever wondered which are the best cities for senior living? I have. And I think that a city that is wheelchair-accessible will be in high demand for seniors who want to relocate. The top 10 cities would be Washington, Seattle, Los Angeles, Denver, Portland, San Francisco, Las Vegas, Orlando, Philadelphia, and Chicago. Plan to move in one of them by calling a real estate agent today. If you already live in one of these cities, then you're fortunate! All you have to do is to rethink your home's interior.
Now is the time to prepare yourself for your future. Plan a renovation and fast forward to the age of say 65: what will you need? You'll need wider doors, at least a full bathroom and a bedroom on the first floor, bathroom improvements like a shower seat and grab bars. In the kitchen, install a pull-out pantry and a shallow sink, and opt for drawers instead of doors. And also, build ramps, install a stair lift or an elevator, if you can afford it! Chair lifts will also help you get up easier.
But another fantastic invention is the ceiling track lift. It costs well over $10,000, but such a system is making life easier for those who have to take care of a person with reduced mobility. There are a few manufacturers on the market, and they sure know to put a price on movement. But does it really matter how much you spend to improve the quality of life of your child, or a loved one with special needs?
In 2013, the Centers for Disease Control and Prevention (CDC) had highlighted mobility as crucial for the elderly, that's why you should take good care of your articulations from the first signs of arthritis. But there are more causes for impaired mobility in seniors like fibromyalgia, neuromuscular conditions, Parkinson's disease, and osteoporosis. Persons with disabilities should also stick with CDC for more information and advice on how to prevent life-threatening diseases or a second disability.
Every time I see Jane and Ania I keep thinking about what could happen if Jane gets a disability after a certain age. How will she care for Ania? She does have a father… but will he cope? Then I remembered that there are also adult foster homes and group homes for adults – both forms of residential communities for adults with special needs – where Ania can live a full life, and where other people with disabilities will comfort her when her mother is no longer next by her side. Jane has good friends, and she is attending a church every Sunday, so there are plenty of people who know their story and who can help her unconditionally. I think Jane did a good job creating a social network underneath her disabled child. In this way, if anything happens to her, Ania won't fall to the ground but will continue to get all the help she deserves.
As children with disabilities get older, their parents have to deal with their own health problems as well. They might soon need special care themselves. In this case, a Long Term Care Insurance is vital, since it protects the child and will guarantee long-term care. Usually, children care for their parents, but you can't expect this from those with special needs. Parents look after their children up to a point. Then, there are the state programs and the private programs.
Parents with disabled children must take legal measures to name a "decision maker" not only for their own health and financial decisions, but for their child/children, too. Choose also a person you trust to sign a power of attorney (POA). This is a document that empowers this person to make decisions regarding your money and your assets. In case you become unconscious, nobody can access your bank account without your permission, not even your child! In the worst scenario you can go to court and get a conservatorship. A healthcare proxy works like the power of attorney regarding your future medical care – this means you empower someone to decide whether to put you on life support or not, or whether to accept a certain surgery or not.
It is very unlikely that a person with special needs will grow old at home. Assisted living and skilled nursing facilities are better places to spend the last years of life. We can expect to see a high demand for special needs housing in the years to come. All these vulnerable people will need more and more help and around-the-clock assistance. Who will take Ania out of the house when her parents will be in a wheelchair too? Of course, she might not live so long, but what if she is the one to prove all statistics wrong?
Special needs housing should provide a certain level of independence to its residents as well as full-time care and assistance, like at Friedman Place, Chicago. They have been providing housing, programs, and services for 81 years. This place is a community for adults who are blind or visually impaired, but this concept should be borrowed immediately to give people with other disabilities a brighter future. Once again, we can see that what people with disabilities value more is not the exterior of the house, but the interior and what happens inside – the love, the care, the respect they get and finding the community that accepts them and makes them feel useful.


I understand the tax laws better than almost anyone. And that is why I am one that can truly fix them. I understand it, I get it. That is what I commit to do. With fairness. We want money brought in, and we want money to be spent when it goes out. Because they spent our tax dollars so unfairly and unwisely. Remember that.
As a businessman and real estate developer, I have legally used the tax laws to my benefit. And to the benefit of my company. I have brilliantly used those laws. On the campaign trail I have often said that I have a fiduciary responsibility to pay no more taxes then is legally required. Put another way, to pay as little tax is legally possible.
I don't know what he means at the end there about fiduciary responsibilities. He has a fiduciary duty to his investors to maximize profits by paying the minimum possible corporate income taxes, but the returns obtained by the Times were his personal returns. He can do whatever he likes with his individual tax liability. He could have taken the loss in 1995, paid no tax that year, but then declined to carry the losses forward once his wealth had rebounded. (And he may have done just that. We'd know if he'd release his returns, as 73 percent of the public wants him to do.) The fiduciary stuff here is a red herring, presumably designed to confuse people into thinking he was under some legal obligation to minimize his own personal tax burden when he really wasn't. But never mind that: The rest of it is straight "subverting the rigged system from the inside" populist spin, with Trump in the role of the master criminal hired by the police to solve murders because he supposedly understands the criminal mind better than anyone. His base will be fine with it, I assume. How about undecideds?
If you read the full transcript of what he said, you'll find that he went on to note that while his 1995 $900 million loss may seem steep, he rebounded to become the yuuuuge financial success he is today. That's much better spin than sending Giuliani and Christie out there yesterday babbling about what a "genius" he was to have avoided tax liability in a year that he, um, lost nearly a billion dollars. Here's Trump from today's rally, followed by Clinton knifing him predictably for having not paid his fair share. Exit question: Is this really a guy who should be lecturing about fiduciary duties? He's not known for putting his shareholders' welfare above his own.


text:
Driver Flees After Hitting, Killing Woman On Redondo Beach Street
Filed Under:Redondo Beach
REDONDO BEACH (CBSLA) – A woman was killed and a man was injured after being struck by a hit-and-run driver in Redondo Beach early Monday morning.
The pedestrians were in the westbound lanes in the 1900 block of Artesia Boulevard when they were struck by the motorist at about 2 a.m., said Redondo Beach police Lt. Shawn Freeman.
(CBS2)
It is unclear if the victims were in a crosswalk.
It was not immediately known if the woman died at the scene or at an area hospital. The man was transported to an area hospital, where his condition was not immediately known, Freeman said.
The motorist was last seen driving away to the west on Artesia Boulevard, the police lieutenant said.
Redondo Beach police traffic investigators asked anyone who saw the crash to call them at 310-379-2477.
(© Copyright 2018 CBS Broadcasting Inc. All Rights Reserved. City News Service contributed to this report.)

redpajama_set_name: RedPajamaCommonCrawl


I opened the card and reread my 16-year-old daughter's scrawled note in the dim light: "I'm happy you care this much about something that will affect the future of us all," she wrote. "Good luck!" I smiled—imagining her brown eyes and flaming red hair—and turned off the light again.
It's not something they warned me about in What to Expect When You're Expecting, but for me, a fierce desire to protect the world my children will inherit is part of being a mother. Along with the sleepless nights I spent rocking my babies when they had a fever, I now count the two times I've gone to jail as part of the sacrifice I willingly make.
It would be nice if our politicians made protecting our children their first priority, but it's become clear that they don't. Extreme methods of extracting fossil fuels—which can cause cancer, birth defects, and other health problems—are profitable and strongly promoted by the oil, coal, and gas companies. Politicians are way more likely to say no to those methods when ordinary citizens stand up and demand that they put people over profits. That's why I handcuffed my wrist to the White House fence to stop the Keystone XL Pipeline, along with 47 others, including the father-son team of Robert Kennedy Jr. and his son Conor. Since then, I've met many other parents whose work to keep the planet habitable is motivated by their children, which I believe is closer to the real history of Mother's Day than brunch or flowers.
Both Mother's Work Day and the Mother's Day Proclamation urged women to work, not just for their own children, but for every child. That sense of solidarity with other mothers is part of what motivates me, too. A former Peace Corps volunteer in Botswana, I've been saddened to learn of how maize crops in Africa are already withering due to rising temperatures and changing rain patterns. I think of my old neighbor, Mmadithapelo, and can't help wondering what the future will be like for her son and daughter, Mopati and Thsego, who are only a few years older than my son and daughter. The temperature in Botswana already gets up to 113 degrees way more often than it did when I lived there in the '80s. I know my own children are not the ones most at risk.
A lot of motherhood is about doing our best and hoping it makes a difference. We might not really know the effect of our actions for a few decades, though there are moments that we get a glimmer that our sacrifices are not for naught. The encouraging note from my daughter the night before my first arrest was one of those moments. It made me want to reach out like that polar bear—and like the early leaders of Mother's Day—and take a leap.
Eileen Flanagan is the author of three books, most recently, "Renewable: One Woman's Search for Simplicity, Faithfulness, and Hope," which is the story of her midlife journey to climate change activism. Now the board chair of Earth Quaker Action Team, you can follow her on Twitter or reach her through her website eileenflanagan.com.


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Ben, Reid, Lea, Oliver & Jack talk to ESLstarter about their experiences teaching English in Thailand. Read on for more first-hand insight and tips about living and working in South East Asia!
Where in Thailand was your placement and which age group did you teach?
I lived in Kantharalak, Sisaket in north-eastern Thailand (an area that is referred to as Isan in Thailand). Isan is a beautiful rural area between Cambodia and Laos full of historic monuments and national parks. My town, Kantharalak, is a small friendly town, which had everything I wanted, even a Tesco store! At my placement school, Kantharalak Wittaya School, I taught students who were13-16 years oldin Mathayoms 1, 2, and 3.
Why did you decide to teach in Thailand?
I lived in China during my University studies and after I graduated I wanted an opportunity to travel and live in another culture. I chose Thailand because it is a beautiful, interesting country, and very culturally different from much of Asia. As the only Asian country not to be colonized, Thailand retains a constitutional monarchy, and a rich cultural heritage. This rich culture (and the amazing cuisine) attracted me to teach in Thailand. In addition, I wanted to use my abilities as a native speaker and a teacher to help enrich the lives of the Thai students.
What was the best thing about teaching in Thailand?
One of the best parts about teaching was seeing the impact I made in Thailand. I saw many students transform from shy children who could not utter a full sentence in English to confident English speakers. The best moment for me was watching my Mathayom 2 class give full Powerpoint presentations and speeches in English, utilizing the language they learned in class!
What were the biggest challenges you faced when teaching in Thailand?
My greatest challenge was learning enough of the Thai language to feel comfortable in my new life. After months of practice, I gained enough language skill to speak to many of the people I encountered on a daily basis. Furthermore, the people I interacted with seemed to greatly appreciate that I tried to learn the language, which gave me a richer living experience in my town.
What advice would you give to first time teachers in Thailand?
Be accommodating and flexible. Thailand will be a new experience and culture shock is natural. Have an open mind about everything that you encounter in and out of the workplace. Be accommodating to your co-workers and the school and remember that you are a guest at the school and in the country.
How long did you teach in Thailand and what are you doing now?
I taught in Thailand for one full school term (six months) and enjoyed it immensely. In fact, I enjoyed teaching so much that I am now applying to teach in Korea!
Where were your favourite places to visit in Thailand / South East Asia?
My favorite place in Thailand was Kanchanaburi. It is a very friendly city with a lot to keep you busy. You can rent motor bikes, ride elephants, visit ancient ruins, and hike through beautiful national parks filled with wildlife and waterfalls. Do not forget to visit the night market, I ate some of the best food in Thailand there!
My other favorite place in Thailand was Chiang Mai, Thailand's second city. It is a spacious and laid back city that was almost impossible to leave. Chiang Mai has a rich local culture and the food was incredible. Just south of the city was Doi Inthanon, the tallest mountain in Thailand and a beautiful national park.
I was placed in Nan. It's in the Northern region of Thailand, near the Laos border. It was a very enjoyable town. Great food, great people, great weather. I taught M.1-M.6 (8th grade through 12th grade), at Strisrinan school for one semester. I then transferred to Nan Technical College.
I was looking for a good experience and I wanted to travel and see more of the world. Also, I thought it would look really good on a job application later on in life. Also, keep in mind that you're going to be in Thailand... which is a gateway to so many other Southeast Asian countries.
The kids... or the food, haha. The children will make you want to go to school each day. To them, you're more or less a celebrity.
The language barrier, or the stares. I was located ten hours away from Bangkok. The town I was in was very rural, that being said, the amount of white people that travel through Nan is slim... people would sit and stare for minutes at a time. I found that that was hard to handle from the beginning. However, its something you will get used to. Not to mention that if being stared at is the worst part of your day... you're doing pretty well, haha. As far as the language goes, again being so far away from Bangkok limits the amount of English that people speak. Thai is not an easy language to learn, still if you give an effort, they LOVE it.
BE OPEN MINDED. Go in it thinking that you're going to be the best teacher in Thailand, then, when reality hits, realize that it's not possible. Not to be a downer or anything, but you need to keep things like that in mind. The Thai peoplesdesire to learn English is not very high, that'sjust the way it goes. Not to be demeaning to them by any means,it's just the facts.Also, don't take things too serious...true, it's education and it should always be taken serious. But the morefun you have, the more they willlearn.Keep it clear, simple, concise and fun.
I was in Thailand for one full year. I am now working as a manager at a golf course in my hometown, and will soon be applying to gain my teaching certification in the states. Fingers crossed.
I was located in Nonthai, a small village within the Nahkon Ratchasima province. The village may have been small but it was very welcoming and had character in abundance. I used to love waking up to the sounds of the bells of the local temple and walking to school as I watched the village come alive in the early morning. The village was built around the local market - a gathering of social activity. Every evening I used to pick up my evening meal there and would always be welcomed and greeted as if I was a local myself. By being located within rural Thailand, I definitely feel I experienced authentic Thai culture and I encourage any potential teacher to grab a rural opportunity with both hands.
My school was a part of a series of private language schools - there were five of them located within the Nahkon province. I taught kindergarten and all grades up to grade 9. This was such an excellent introduction to the world of teaching for me. I quickly learnt how to create and develop appropriate lesson plans in regards to the age and ability of my students. I absolutely adored teaching kindergarten. Every morning, I would be greeted by a sea of beaming faces, pleased to see me and to show me their latest colouring creations. In contrast, I also enjoyed teaching the later grades as I could converse with them, learn about them as young people and help to teach them the relevant English skills that they both wanted and needed to know. It was such a privilege to meet such smart students and to learn of their dreams, goals and ambitions.
This was my first serious teaching job and I received assistance from teachers at all times, meaning I could learn and develop at my own speed.
First of all, let me tell you a little about myself. After graduating from Uni, I found myself unsure of where to go next. My life was at a crossroads, with a lot of my time devoted to applying to meaningless and faceless graduates schemes. After reading ESLstarter's advertisement, I was amazed. It was such a good opportunity. They were a company that actively recruited novices, provide them with TEFL training / award and secure them an job within the country of the applicant's choice. In particular, Claire was instrumental in swaying me to pursue an ESL career. She handled all of the tricky visa / employment paperwork and provided valuable assistance to whenever I had any queries. All in all, I've been given a top service and I wouldn't be employed now here in China if it wasn't for ESLstarter.
I picked Thailand for plethora of reasons. First of all, who wouldn't want to live and work in such a beautiful country? The chance to leave behind rainy England and pursue a new adventure in the land of a million smiles! Then there is the food. You'll be amazed at the cooking and how varied it is. As mentioned, every evening I used to go to the market to pick up my evening meal. I used to love buying food from the stalls - I could buy fried chicken, Thai omelette, rice and a bottle of beer...and still have change left over! The food is very quick, very delicious and it's so easy to eat healthy! I also picked Thailand as an opportunity to experience the spiritual side of Buddhism. Over my teaching contract, I was lucky enough to attend Buddhist gatherings in the local temple and join in myself with the ritual. One time, I was also lucky enough to be taken to the meet the head monk. That was such a privilege, one I certainly won't forget!
Finally, I picked Thailand because I knew the opportunity would be whatever I made of it. You can go to Thailand and participate in the tourist activities, such as full moon parties. Or you can do what I did - leave behind urban Thailand and head to the country to witness a culture free from Western strains. Whatever you do, it's your opportunity to make fabulous life experiences and I wish you the best of success.
It's so hard to narrow down to just one example. I can't as there are many faces to Thailand and so many experiences I treasure. I would have to start by mentioning my wonderful principal. This kind woman did so much for me and I can never pay her back enough. She took care of me whenever I felt ill or lonely, she assured me whenever I felt like a rotten teacher, and she helped to make me feel welcome in my new home and community. Not only was she my friend, she was also very motherly towards me and would often refer to me as her second son. I am still in touch with the principal and I plan to return to my school in January to see her. I can't wait. One of my most endearing memories of my principal is the time we celebrated Loi Krathong together. The day itself brought bad weather and we were unsure whether the full moon would be visible. Thankfully the celebration took place, and both myself and the principal made our way to the river. As I floated my Krathong into the water, I looked up and I saw lanterns floating their way across the night sky, being watched by the moon itself. I looked back at the water to see numerous Krathongs floating with their flames delicately dancing, representing the hopes, dreams and ambitions of the community that had gathered by the river to celebrate. I stood with my principal and just drunk it in. It was magical.
I have been both lucky and blessed to have worked with a wonderful principal and set of teachers. The teachers were so good to me. They loved to practice their English with me, would often give me a lift to the shops, and often invited me to their celebrations. By far, the most touching memory that I have is the time I attended a teacher's wedding. I have a lot of affection for this teacher. He was a great guy and we'd spend hours talking about football - I even played for the local teacher team...we got beat 4-1! Anyway, it was three weeks after I had left school to travel Thailand when I received an invite to his wedding (via Facebook). Immediately I dropped all of my plans and travelled back to the community. It was a wonderful party and I was so touched to be invited. You'll never be alone in Thailand as you will have the support and friendship of your new teacher colleagues.
Oh, I haven't even mentioned the kids yet! The kids were so happy and pleased to meet me. In Thailand, the role of teacher is greatly revered. It commands degrees of respect and esteem within the local community. Upon this, the kids were excellent towards me. They had such a strong work ethic and a desire to work for their futures. The kids will love to speak English with you! Although it might be tempting to try and speak Thai to them outside the classroom, keep speaking English as they love to practice, especially in front of their parents. It's that moment when you see a proud parent beaming at their child that makes you realise just why you wanted to be a teacher!
One thing that I appreciated about teaching in Thailand is the freedom that I was given towards teaching the lessons that I wanted to teach. You are free to create your lessons and deliver them the way you want. Make them fun, make the kids laugh and help them to speak English. My personal ambition was to increase conversational skills. Rather than drill and make grammar exercises, I wanted the children to use their initiative and logic to turn English into a dynamic language of art and expression.
There were so many opportunities for me to get involved within the Thai way of life. I hope you make the most of them as you will make friends for life.
There were definitely periods where I felt so alienated within the Thai community. Sometimes I felt so out of place and I also was frequently frustrated with the language barriers - Thai is very difficult! It felt like I was restricted in regards, as if I didn't have the freedom to move about with ease. Living within a small village, I also found myself missing home and friends a lot. It's only natural and a lot of teachers experience degrees of alienation. However, it is also very easy to overcome. If you find the language difficult, use it as an incentive to call a teacher to help you practice. That way you can socialise better and be seen to be actively involving yourself within the community. If you find yourself frustrated with your surroundings, hop on a bus and just see where it takes you! Stay on until the end of the line and sight-see a little! You never know what you might find!
As a foreigner in rural Thailand, you will find yourself to be an attraction to the locals, maybe because they rarely get the opportunity to meet and greet foreigners. Certainly, I was often stared at by the locals. Sometimes it felt so strange but please don't mistake it for malice. They are genuinely excited and honoured that you're a teacher in their community, even they are just staring blatantly at you with their mouths open!
The village also had numerous stray dogs who would love to spend their time chasing me! It is a bit of an issue, especially if you're afraid of dogs, but hey at least I got to school a little earlier.
Always keep an open mind. Thailand is equally frustrating and rewarding but it needs your toleration and appreciation. If you are prepared to devote time and effort to undertaking the culture, you will be loved. If you find yourself frustrated, just remind yourself that it's a different culture - it's not right and it's not wrong. It's just different. You can't change it so keep smiling and keep patient with your opportunity, and you will be rewarded with affection and kindness.
When you arrive at your new school, buy a small pocket book and ask an English-speaking teacher to write down key phrases in Thai. That way, if you struggling to communicate with a Thai person, you can just open the book and show him / her the phrase in Thai. That's particularly useful if you have food allergies and want to know if certain foods contain nuts and so on.
Start to think about your teaching ideas. Think about how you can introduce your culture to Thai children, and how you can introduce them to their own culture through the means of the English language. Make your lessons as full and as engaging as possible. You are the teacher, it is up to you! The more you teach, the more you develop a magpie-like tendency of mentally gathering up and storing good ideas for your children, even if you're not aware of it! You will find that the lessons soon write themselves!
I taught in Thailand for 6 months and now am working at a private language school in China.
In a heartbeat, I would say Ayutthaya. I adore that place. So many temple ruins, good restaurants and sights to see. They also have the sweetest lemonade there too. Ayutthaya was once the capital of Thailand. Go there and you'll feel an overwhelming sense of history and tradition as you marvel at the ruins of the wonderful city.
I was placed in Satun Province in the south of Thailand teaching students aged 12-15.
For me it was the thrill of the unknown as well as wanting to experience the white beaches of Samui, the bustle of Bangkok and the opportunity to experience a completely different culture.
The holidays. The buckets. The people. The beaches. Khan San Road. If I had to choose just one though, it would be the students. They make the job fun and easy!
The language barrier. In Satun, most people can say 'hello, how are you?' But that's it. It was a real struggle at first trying to ask basic questions and to get around the town.
'Mai Pen rai' (no problem) and 'why so serious?' are two of the most said phrases. Just relax and smile - they'll love you straight away!
I taught for a year at Princess Chulabhorn's College, Satun and have since moved back to London to start life as a banker in the city.
Nothing beats a crazy weekend in Khao San Road, Bangkok. Other highlights were the Full Moon Party on Koh Phangan, Songkran Festival in Chiang Mai and Vietnam. If you get a chance to travel to other countries around Thailand do it! Each one is very different to the last.
Where in Thailand is your placement and which age group do you teach?
I'm currently in Ubon Ratchathani, which is up in the northeast. I teach students from M1 to M4, which is about the equivalent of years 7-10 in the UK.
There were a few different reasons, but mainly it was the fact that I wanted to travel and go abroad but didn't have the money to do it without working. I'd been thinking about different options and just decided to try it. Out of the different areas where teaching was possible, Thailand really appealed to me - i'd heard good things from people who had been there and it's famous for food and being a fairly laid-back place, both of which were things I was looking out for. Plus, the cost of living here is pretty low so it was a good way to make the most out of the little money I had lying around at the time.
What is the best thing about teaching in Thailand?
Teaching here really gives you an opportunity to do and see a lot of things you wouldn't get to otherwise. Of course you can get a feel for a place when you're just passing through, but when you live somewhere for a few months you can get a proper idea of what it's about. As a teacher you can be a part of the community, especially if you end up in a small town in the middle of nowhere. It gives you the chance to go places and see things with the people you meet, and access to so much stuff you simply wouldn't come across on a holiday. As a general rule, I have found Thai people to be really friendly and accommodating, and if you're game to take part in things when they invite you then it can be great fun. There are some really stunning places around outside of the main tourist trails and some fantastic experiences to be had here, and teaching is a passport to get involved in it. You can do a lot worse in terms of the actual work, too.
What are the biggest challenges you face when teaching in Thailand?
General disorganisation, which can be extremely frustrating, especially when you get activities or obligations sprung on you with anything from two days to literally minutes notice. The occasionally terrible internet. Also, sweat patches, which are a constant menace.
Be flexible. It can be extremely frustrating to begin with because things just don't run the same way here, and the basic expectations you might have of working somewhere in the west sometimes simply don't apply. The angriest and unhappiest foreign teachers I've seen here have been the ones who have constantly tried to judge everything by the standards they'd apply back home, and have fought against the Thai way of doing things, often misunderstanding or willfully ignoring the way people around them will feel or react. Sometimes you just have to accept that there's a different kind of logic to your own being applied and be prepared to adapt to that - and once you do, everything just becomes a lot smoother. Also, don't bring too much stuff; you can buy everything you need here, especially in Bangkok. I chanced it with just hand luggage coming over and was genuinely amazed how much easier it made things. A Kindle or something is a good idea too if you want to read because finding books in English is tricky in Bangkok and nearly impossible outside of it.
How long have you been teaching in Thailand so far and how long do you plan on staying?
I've been here for just over 14 months now, although it really doesn't seem like that long. My contract runs until the end of April, but as of now I have absolutely no idea whether I will renew it and stay here or bugger off somewhere else. I definitely wouldn't have a problem with staying here for the same amount of time again, if not longer.
Where are your favourite places to visit in Thailand / South East Asia?
My favourite places so far have probably been Koh Tao for the awesome combo of nightlife, diving and mountains, Koh Samet for riding dirtbikes and not being too full of people in fluorescent vests, and Khon Khaen for just being a beautiful city and genuinely nice place to be. Also, Phra Taem national park near the Laos border has some spectacular waterfalls to splash about in and is the place to watch the first sunrise in Thailand from on top of a cliff, with ridiculously pretty views out over the Mekong river and Laos.
Check out ESLstarter's Explore Thailand Program!


We are a producer of windows and doors from all imported wood Manufacturer& Exporter. We have been looking for importers worldwide contact.... We are in the timber since last few decades. Our operations are being carried out in all over India. We do have branches other parts of the country. Our company is engaged in sawmills, wholesale distribution of logs and sawn sizes. Just now we deal in Malaysian& Indonesian hardwood along with TEAK from Africa& Burma. We also make pallets and wooden boxes of Pine and Indian Jungle Hardwood. As we are the third generation in the same business we are looking forward for the expansion and huge business.


text:
\section{Introduction}

The theory of compressive sensing predicts that sparse vectors can be stably reconstructed from a small number of linear measurements
via efficient reconstruction algorithms including $\ell_1$-minimization \cite{do06-2,fora11}. The restricted isometry property (RIP) of the measurement matrix streamlines
the analysis of various reconstruction algorithms \cite{cata06,carota06-1,fo10,fora13}. All
known matrices that satisfy the RIP in the optimal parameter regime (see below for details) are based on randomness. Well-known
examples include Gaussian and Bernoulli matrices where all entries are independent. Unfortunately, such matrices do not
possess any structure and therefore no fast matrix-vector multiplication algorithm.
The latter is important for speed-up of recovery algorithms. This article addresses constructions of
matrices that satisfy the RIP in the optimal parameter regime and have fast matrix-vector multiplication algorithms.

A vector $x \in {\mathbb C}^n$ is said to be $s$-sparse if the number of nonzero entries of $x$ is at most $s$.
A matrix $A \in {\mathbb C}^{k \times n}$ satisfies the RIP
with respect to parameters $(s,\delta)$ if, for all $s$-sparse vectors $x\in {\mathbb C}^n$,
\begin{equation}\label{basic}
(1-\delta)\|x\|_2 \leq \|Ax\|_2 \leq (1+\delta)\|x\|_2,
\end{equation}
where $\|\cdot\|_2$ denotes the Euclidean norm.\footnote{In much of the related literature, the definition of RIP uses \emph{squared} Euclidean norms.  The definition (\ref{basic}) is, however, more convenient for our purposes. Of course, both versions are equivalent up to a transformation
of the parameter $\delta$.}
\na{(I moved the footnote from inside the sentence to after the period - I think it's the correct way to do so in English, at least that's what some anoynous pedantic reviewer told me once)}
If $A$ satisfies the RIP with parameters $(2s, \delta^* )$ for a suitable $\delta^* < 1$ then a variety of recovery algorithms reconstruct
an $s$-sparse vector exactly from $y = A x$. Moreover, reconstruction is stable under passing from sparse to approximately sparse vectors
and under adding noise on the measurements. The value of $\delta^*\xspace $ depends
only on the reconstruction algorithm \cite{carota06-1,netr08,blda09,fo10,fora13}.

It is well-known by now \cite{badadewa08,cata06,mepato09} that a
Gaussian random matrix (having independent normal distributed entries of variance $1/m$)
satisfies the RIP with parameters $(s,\delta)$ with probability at least $1-e^{-c \delta^2 k}$ if
\[
k \geq C \delta^{-2} s \operatorname{log}(n/s),
\]
where $c,C>0$ are universal constants. Using lower bounds for Gelfand widths  of $\ell_p$-balls for $0 < p \leq 1$,
it can be shown that $k$ must be at least $C_{\delta} s\operatorname{log}(n/s)$ for
the RIP to hold \cite{FoucartPRU10}.
It can further be shown \cite{fora13} that the constant (as a function of $\delta$) satisfies
$C_{\delta} \geq C \delta^{-2}$.
Since we will always assume in this paper that
$s \leq Cn^{1/2}$, $\operatorname{log}(n/s)$ is equivalent to $\operatorname{log}(n)$ up to constants.
Hence, we will say that a $k \times n$ matrix is {\bf RIP-optimal} at $s$
if it satisfies the RIP with $(s,\delta)$ for
\[
\delta = C \sqrt{\frac k {s\operatorname{log} n}}\ .
\]
 (The reader should keep in mind that for large $s$,
RIP optimality should be defined to hold when $k$ is at most $C\delta^{-2}s\operatorname{log}(n/s)$.)

The restricted isometry property is closely connected to Johnson-Linden\-strauss embeddings.
We say that a random $k \times n$ matrix $A$ satisfies the Johnson-Lindenstrauss  property (JLP) with parameters
$(N,\delta)$ if for any set $X \subseteq \mathbb R^n$ of cardinality at most
$N$  (\ref{basic}) holds uniformly for all $x\in X$ with constant probability.
It is well known that a matrix of independently drawn subgaussian elements satisfies JLP if $k \geq C\delta^{-2}\operatorname{log} N$.
Specializations of this fact
to Gaussians and to Bernoulli random variables can be found in \cite{Indyk:1998:ANN:276698.276876,DBLP:journals/jcss/Achlioptas03}. The general claim is  obtainable by noting that the subgaussian
property is the crux of these proofs.  If $A$ satisfies JLP with $(N,\delta)$ for $k \leq C\delta^{-2}\operatorname{log} N$, then we say that $A$ is JLP-optimal.

The JLP and RIP properties are known to be almost equivalent, in a certain sense.
One direction is stated as follows: A (random) $k \times n$  matrix satisfying JLP
with $(N,\delta)$ satisfies
RIP with $(\hr{c}C(\operatorname{log} N)/(\operatorname{log} n), \delta)$ \na{(agreed)} with constant probability \cite{badadewa08,mepato09}.
Hence, for any arbitrarily small $\mu>0$ an RIP-optimal matrix can be obtained by drawing a  JLP-optimal matrix with $k=C \delta^{-2}s \operatorname{log} n$.
The derivation of RIP from JLP is a specialization of JLP to a set $X$ consisting of a $\varepsilon$-net of $s$ sparse unit vectors, for $\varepsilon = 0.1$ say, which has cardinality
$N \leq (C n)^s$.

The other direction is a remarkable recent result by Krahmer and Ward \cite{DBLP:journals/siamma/KrahmerW11} implying that if $A$ has RIP with $(s,\delta/4)$,
then $AD$ has JLP with $(N, \delta)$ as long as $N \leq 2^s$, where $D$ is a diagonal matrix with independent random signs ($\pm 1$) on the diagonal.  Notice that from this
result, RIP-optimality of $A$ does not imply JLP-optimality of $AD$, because RIP-optimality implies
that the embedding dimension of $k$ is at least $C\delta^{-2} s\operatorname{log} n$, which suffers from an additional factor of $\operatorname{log} n$ compared to the JLP-optimality guarantee bound
of $C\delta^{-2}\operatorname{log} N = C\delta^{-2}s$  (for $N=2^s$).  From this observation we intuitively conclude that RIP-optimality is weaker than JLP-optimality,
and hence expect that RIP-optimal constructions should be easier to obtain.  The main results of this paper, roughly speaking, confirm this by providing constructions of RIP-optimal matrices
which are simpler than previously known constructions\na{\cite{DBLP:journals/dcg/AilonL09,aich09}} \hr{WHICH ONES?}that relied on JLP optimality.

\subsection{Known Constructions of RIP-optimal and JLP-optimal matrices}

No deterministic RIP-optimal matrix constructions are known.  Deterministic constructions of RIP matrices are known only for a grossly suboptimal regime of parameters.
See \cite{DBLP:journals/siamma/KrahmerW11} for a nice survey of such constructions.

Of particular interest are RIP or JLP matrices $A$ that are \emph{efficient} in the sense that for any vector $x\in {\mathbb C}^n$, $Ax$ can be computed in time $O(n\operatorname{log} n)$.
Such constructions are known for JLP-optimal (and hence also RIP-optimal) matrices as long as $k \leq n^{1/2-\mu}$ for any arbitrarily small $\mu$ \cite{DBLP:journals/dcg/AilonL09}.
This is achieved with the transformation $BHD^{(1)}HD^{(2)}H\cdots HD^{(r)}$,
where $D^{(i)}$ are independent random sign diagonal matrices, $H$ are Hadamard transforms
and $B$ is a subsampled (and rescaled) Hadamard transform, where the subset
of sampled coordinates is related to a carefully constructed dual binary code, and $r$ is at most $C/\mu$.
For larger $k$,
the best efficient constructions satisfying RIP are due to Rudelson and Vershynin \cite{ruve08}
with $k \geq Cs\operatorname{log}^4 n$, namely  a factor of $\operatorname{log}^3 n$ away from optimal, see also \cite{ra10}. (Note that
at least two of the $\operatorname{log} n$ factors can be improved to $\operatorname{log} s$, but in light of
the aforementioned positive results, we can assume that $s$ is at least, say, $n^{1/3}$.)
This was recently improved by Nelson et al. to only $\operatorname{log}^2 n$ factors away from optimal \cite{NelsonPW12}.
The construction in \cite{ruve08} is a Fourier transform
followed by a subsampling
operator.
Another family of RIP almost optimal matrices with a fast transform algorithm is that of partial random circulant matrices and time-frequency structured random matrices.
The best known results in that vein have recently appeared in the work of Krahmer, Mendelson and Rauhut \cite{KrahmerMR12}, where RIP matrices of almost optimal embedding
dimension $k= Cs\operatorname{log}^4 n$ are designed.
These constructions improve on previous work in \cite{rarotr12}.

\subsection{Contribution}
In this work we construct RIP-optimal, efficient matrices for the regime $s\leq n^{1/2-\mu}$ that are simpler than those implied from the JLP construction in \cite{DBLP:journals/dcg/AilonL09}.
For $s\leq C n^{1/3}/\operatorname{log}^{2/3} n$, we show that the transformation $SHDHD'H$ is RIP-optimal, where $H$ is a Hadamard or Fourier transform, $D$ and $D'$ are independent  random sign diagonal matrices
and $S$ is an \emph{arbitrary} deterministic subsampling (and rescaling) matrix of order $k$.  This complements a previous RIP result in a similar parameter regime of $s \leq n^{1/3}/\operatorname{polylog}(n)$ \hr{(Should we be more specific here? Danger of confusion between the two regimes just mentioned!)} \na{done}
implied by a JLP construction by Ailon and Chazelle in \cite{aich09},
in which the transformation $THD$ is used, where $T$ is a sparse random matrix with $k^3$ nonzero elements.
 The surprising part about our construction is that
$S$ can be taken to
be an arbitrary (deterministic) subsampling matrix.  The random subsampling used in the construction \cite{aich09} is, in a sense, traded off with two more Fourier transforms and one more
random diagonal matrix here.
For $s\leq Cn^{1/2-\mu}$, we show that the transformation $PD^{(1)}HD^{(2)}H\cdots D^{(r)} H$
is RIP-optimal and efficient, where the $D^{(i)}$'s are as above, $P$ is an \emph{arbitrary} deterministic matrix with properly normalized, pairwise orthogonal rows and $r$ is at most $C/\mu$.
This is simpler than the  RIP-optimal efficient construction implied by the aforementioned  JLP-optimal efficient construction in \cite{DBLP:journals/dcg/AilonL09}, because no binary code designs are necessary.

Our main proof techniques involve concentration inequalities of vector valued Rademacher chaos of degree $2$. For the second construction an additional bootstrapping argument is applied
that, roughly speaking, shows that the RIP parameters of $ADHD'H$ are better than those of $A$.

\section{Notation and Main Results}

Throughout, the letter $C$ denotes a general global constant, whose value may change from appearance to appearance.
The integer $n$ denotes the ambient dimension, $k \leq n$ denotes the embedding dimension, and
${\mathbb C}^n$ denotes the $n$ dimensional
complex space with standard inner product. The usual $\ell_p$-norms are denoted by $\|\cdot\|_p$, the spectral norm of
a matrix $A$ by $\|A\|$ and the Frobenius norm as $\|A\|_F = \sqrt{\operatorname{trace}(A^*\xspace A)}$.

\noindent
We let $H\in {\mathbb C}^{n\times n}$ denote a fixed matrix with the following properties:
\begin{enumerate}
\item $H$ is unitary,
\item the maximal absolute entry of $H$ is $K{K}n^{-1/2}$K{, for some number $K>0$ which we assume fixed},
\item the transformation $Hx$ given a vector $x\in {\mathbb C}^n$ can be computed in time $O(n\operatorname{log} n)$.
\end{enumerate}
 K{The discrete Fourier matrix  is an example of such matrix (with $K=1$), and the discrete cosine transform is another (with $K=\sqrt 2$).
The Walsh-Hadamard matrix (for $n=2^\ell$) is another such (real valued) example.}
\noK{Both the discrete Fourier matrix and the Walsh-Hadamard matrix are examples of such matrices.}
\noK{Note that the upper bound of $n^{-1/2}$ in property 2. above could be replaced by $Kn^{-1/2}$ for any constant
$K$, thus encompassing transformations such as the discrete cosine transform (with $K=\sqrt 2$) with little
affect on the guarantees.  We have
decided to concentrate on the case $K=1$ for simplicity.}

For any vector $z\in {\mathbb C}^n$, $D_z$ denotes a diagonal
matrix with the elements of $z$ on the diagonal.
For a subset $\Omega$ of $\{1,\dots, n\}$, let  $\operatorname{ind}(\Omega)\in \mathbb R^n$ denote the vector with $1$ at coordinates $i\in \Omega$ and $0$ elsewhere.  Then define $P_\Omega = D_{\operatorname{ind}(\Omega)}$ and
$R_\Omega \in \mathbb R^{k \times n}$ to be the map that restricts a vector in $\mathbb R^n$ to its entries in $\Omega$, i.e., a subsampling operator.

Recall that a $k \times n$ matrix $A$ has the RIP property with respect to parameters $(s,\delta)$ where $s$ is an integer and $\delta>0$, if for any $s$-sparse unit vector $x\in {\mathbb C}^n$,
$$ 1-\delta  \leq \|Ax\|_2 \leq 1+\delta\ .$$
(Note that we allow $\delta>1$, unlike typical definitions of RIP).
For a fixed sparsity parameter $s$, we denote by $\delta_s(A)$
the infimum over all $\delta$ such that $A$ has the RIP property
with respect to parameters $(s,\delta)$.   We say that $A$
is RIP optimal at a given sparsity parameter $s$ if
\begin{equation}\label{defopt} \delta_s(A) \leq C \sqrt{\frac{s\operatorname{log} n} k}\ .
\end{equation}

A random vector $\epsilon$ with independent entries that take the values $\pm 1$ with equal probability is called a Rademacher vector.

Our first main result provides a simple RIP-optimal matrix with a fast transform algorithm for small sparsities $s = {\cal O}(n^{1/3}/\operatorname{log}^{2/3} n)$.

\begin{thm}\label{thm1}
Let $\epsilon, \epsilon'\in \{\pm 1\}^n$ be two independent Rademacher vectors,
and let $\Omega$ be any subset of $\{1,\dots, n\}$ of size $k$.
The $k \times n$ random  matrix $A = R_\Omega HD_{\epsilon}H D_{\epsilon'} H$
is RIP-optimal
for the regime  $k \leq \sqrt{n/s}$. More precisely, if
\begin{equation}\label{cond:k}
\sqrt{n/s} \geq k \geq CK{K} \delta^{-2} s \operatorname{log} n,
\end{equation}
then $A$ satisfies the RIP \eqref{basic} with probability at least $1-e^{-C\delta^2 k}$.
In particular, the conditions on $k$ entail
\[
s \leq \frac{C\delta^{4/3} n^{1/3}}{K{K^{2/3}}\operatorname{log}^{2/3} n}\ .
\]
\end{thm}
Clearly, $A = R_\Omega HD_{\epsilon}H D_{\epsilon'} H x$ can be computed in ${\cal O}(n \operatorname{log} n)$ time by assumption on $H$.
Also note that \eqref{cond:k} implies a restriction on $\delta$ for which the result applies, namely
\begin{equation}\label{thm1:rest:delta}
\delta \geq c\frac{s^3 \operatorname{log} n}{n}.
\end{equation}

Our second main result gives a RIP-optimal matrix construction with a fast
transform for the enlarged parameter regime $s = O(\sqrt{n}/\operatorname{log} n)$ and $k = O(n/(s\operatorname{log} n))$.

\begin{thm}\label{thm2}
Assume $s_0 \leq s \operatorname{log} n \leq k \leq \sqrt n$ and
$\kappa = C\sqrt{\frac{sk\operatorname{log} n} n} < 1/2$, where $s_0$ is a global constant.
Let $A$ be an arbitrary $k \times n$  matrix satisfying $AA^*\xspace  = \frac n k \operatorname{Id}_k$.
Let $r = \left  \lceil \frac{ -\operatorname{log}(2\sqrt{n/k})}{\operatorname{log} \kappa}\right \rceil$, and
let $\epsilon_{(1)},\dots, \epsilon_{(r)} \in \{\pm 1\}^n$ denote independent Rademacher vectors.
Then the $k \times n$ matrix
\begin{equation}
\hat A = A D_{\epsilon_{(1)}}HD_{\epsilon'_{(1)}} H D_{\epsilon_{(2)}}HD_{\epsilon'_{(2)}} H
\cdots
D_{\epsilon_{(r+1)}}HD_{\epsilon'_{(r+1)}} H
\end{equation}
is RIP-optimal with probability at least $0.99$, that is, \eqref{basic} holds if $k \geq C K{K} \delta^{-2} s \operatorname{log} n$.

In particular, if we strengthen the constraints by requiring $s \leq n^{1/2-\mu}$ for some global $\mu>0$,
then $\hat A$ is also computationally efficient in the sense that $\hat A x$ can be computed in time ${\cal O}(n \operatorname{log} n)$.
\end{thm}

The probability $0.99$ in the above theorem is arbitrary and can be replaced by any different value in $(0,1)$. This effects only
the constant $C$. However, we remark that the present proof does not seem to give the optimal dependence of $C$
in terms of the probability bound.

It is presently not clear whether the restrictions $s = {\cal O}(n^{1/3}/K{(K^{2/3}}\operatorname{log}^{2/3} nK{)})$ and $s = {\cal O}(\sqrt{n}/\operatorname{log} n)$ in the above theorems can be removed.
In any case, regimes of small $s$ are the ones of most interest in compressive sensing, anyway.

Section~\ref{cbrtn} is dedicated to proving the Theorem~\ref{thm1}, and Section~\ref{sqrtn} proves Theorem~\ref{thm2}.

\section{The regime $s = {\cal O}(n^{1/3}/K{(K^{2/3}}\operatorname{log}^{2/3} nK{)})$}
\label{cbrtn}

Our first randomized, computationally efficient, RIP-optimal construction involves three
applications of $H$, two random sign diagonal matrices, and a choice of an \emph{arbitrary}
set of $k$ coordinates.
 Fix  $x\in U_s := \{ x \in {\mathbb C}^n: \|x\|_2 = 1, x \mbox{ is } s\mbox{-sparse} \}$
 and let $\epsilon, \epsilon' \in \{\pm 1\}^n$ be two random sign vectors.
Consider the following random variable, indexed by $x$,
$$ \alpha(x) = \sqrt{\frac n k}\| P_\Omega HD_\epsilon HD_{\epsilon'} Hx\|_2\ ,$$
where $k$ is the cardinality of $\Omega$.
\noindent
It is not hard to see that $\mathbb E[\alpha(x)^2] = 1$.  Indeed, denoting $\tilde x = HD_{\epsilon' }Hx$ and conditioning on a fixed value of $\epsilon'$,  for any $i\in \{1,\dots n\}$,

$$ \mathbb E_\epsilon \|P_{\{i\}} H D_\epsilon \tilde x\|^2_2 = \|\tilde x\|_2^2/n = 1/n\ .$$

The random variable $\alpha(x)$ is the norm of a decoupled Rademacher chaos of
degree $2$.  For the sake of notational convenience, we denote, for $i,j\in \{1,\dots n\}$,
\begin{equation}\label{def:xij}
 {\bf x}_{ij} = \sqrt {\frac n k} P_\Omega H P_{\{i\}} H P_{\{j\}} H x\ ,
 \end{equation}
so that we can conveniently write
\[
\alpha(x) =  \|\sum_{i=1}^n\sum_{j=1}^n \epsilon_i\epsilon'_j {\bf x}_{ij}\|_2.
\]
By a seminal result of Talagrand \cite{Tal96}, a Rademacher chaos concentrates around its median.
We will exploit the following version.
\begin{thm} With a double sequence ${\bf x}_{ij}$, $i,j=1,\hdots n$, of vectors in ${\mathbb C}^n$ and two independent Rademacher
vectors $\epsilon, \epsilon' \in \{\pm 1\}^n$ let
\[
\alpha = \| \sum_{i=1}^n\sum_{j=1}^n \epsilon_i\epsilon'_j {\bf x}_{ij}\|_2.
\]
Let $M_{\alpha}$ be a median of $\alpha$. For
${\bf y} \in {\mathbb C}^n$ introduce the $n \times n$ matrix $B_{\bf y} = ({\bf y}^*\xspace  {\bf x}_{ij})_{i,j=1}^n$ and
the parameters
\begin{align}
U & = \sup_{{\bf y} \in {\mathbb C}^n, \|{\bf y}\|_2 \leq 1} \|B_{{\bf y}}\|\label{defU}\\
V & = \mathbb E \sup_{{\bf y} \in {\mathbb C}^n, \|{\bf y}\|_2 \leq 1} (\|B_{{\bf y}} \epsilon\|_2^2 + \|B_{{\bf y}}^*\xspace  \epsilon'\|_2^2)^{1/2}.\label{defV}
\end{align}
Then, for $t > 0$,
\begin{equation}\label{tal:chaos2}
\Pr(| \alpha - M_{\alpha} | \geq t ) \leq 2 \exp\left(-C \min\left\{\frac{t^2}{V^2},\frac{t}{U}\right\} \right)\ .
\end{equation}
\end{thm}
\begin{proof} With
$A = ({\bf x}_{ij})_{i,j=1}^n$ and
\[
S = \frac{1}{2} \left(\begin{matrix} 0 & A \\ A^*\xspace  & 0\end{matrix}\right), \qquad \tilde{\epsilon} = \left(\begin{matrix} \epsilon \\ \epsilon' \end{matrix} \right)
\]
we can rewrite the decoupled chaos as the coupled symmetric chaos
\[
\tilde{\epsilon}^*\xspace  S \tilde{\epsilon} = \sum_{i,j=1}^n \epsilon_i \epsilon_j' {\bf x}_{ij},
\]
where matrix multiplication is extended in an obvious way to matrices with vector-valued entries. Observe that $S$ has zero diagonal.
Therefore, the claim follows from Theorem~1.2  in \cite{Tal96}.
\end{proof}

We bound the quantity $U=U(x)$ in \eqref{defU},
where the ${\bf x}_{ij}$ are defined by \eqref{def:xij}.
Note that $\sum_{i,j} {\bf \alpha}_i{\bf \beta}_j {\bf y}^*\xspace  {\bf x}_{ij} = {\bf y}^*\xspace  HD_{{\bf \alpha}} H D_{{\bf \beta}} H x$, and hence
\begin{eqnarray}
U & = & \sup_{\|{\bf y}\|_2, \|{\bf \alpha}\|_2, \|{\bf \beta}\|_2\leq 1}  \sum_{i=1}^n \sum_{j=1}^n \bar {\bf \alpha}_i {\bf \beta}_j {\bf y}^*\xspace {\bf x}_{ij}
= \sqrt{\frac n k} \sup_{{\bf y}, {\bf \alpha}, {\bf \beta}} {\bf y}^*\xspace  P_\Omega H D_{{\bf \alpha}^*\xspace} HD_{{\bf \beta}} H x  \nonumber \\
&=& \sqrt{\frac n k}\sup_{{\bf y}, {\bf \alpha}, {\bf \beta}}  {\bf \alpha}^*\xspace  D_{{\bf y}^*\xspace  P_\Omega H} H D_{Hx} {\bf \beta}  \nonumber \\
&\leq&  \sqrt{\frac n k} \sup_{{\bf y}, {\bf \alpha}, {\bf \beta}}\|{\bf \alpha} \|_2 \cdot  \|D_{{\bf y}^*\xspace  P_\Omega
 H}\| \cdot \|D_{Hx}\| \cdot \|{\bf \beta}\|_2 \nonumber\\
&\leq&  \sqrt{\frac n k}  \sup_{{\bf y}, {\bf \beta}} \|{\bf y}^*\xspace  P_\Omega H\|_{\infty} \cdot \|Hx\|_\infty \cdot \|{\bf \beta}\|_2 \nonumber\\
&\leq&  \sqrt{\frac n k} \sup_{{\bf y}} \|{\bf y}^*\xspace  P_{\Omega}\|_1 K{K}n^{-1/2} \cdot \|x\|_1 K{K}n^{-1/2}
\leq  \sqrt{\frac n k} k^{1/2} K{K^2} n^{-1/2} \|x\|_1 n^{-1/2}\nonumber\\
&\leq& K{K^2} \sqrt {s/n} \ . \label{Ubound}
\end{eqnarray}

\noindent
To bound $V$, we define a process $\nu({\bf y})$  as
\begin{equation}\label{process} \nu({\bf y}) = \sqrt{ \|B_{{\bf y}} \epsilon\|_2^2 + \|B_{{\bf y}}^*\xspace  \epsilon'\|_2^2}\,
\end{equation}
so that $V = \mathbb E \sup_{\|{\bf y}\|\leq 1} \nu({\bf y})$.
By the definition of the vectors ${\bf x}_{ij}$, it clearly suffices to take the supremum on vectors ${\bf y}$ supported on $\Omega$.
For any such ${\bf y}$, let $\mu_{\bf y} = \mathbb E \nu({\bf y})$.
Jensen's inequality yields
\begin{equation}\label{boundexpectation111}
\mu_{\nu({\bf y})} \leq \sqrt{\mathbb E \nu_{\bf y}^2} \leq \sqrt {\frac {2n} k} \| D_{{\bf y}^*\xspace  P_\Omega H} H D_{Hx} \|_F\ .
\end{equation}
By definition of the Frobenius norm, together with the fact that the matrix elements of $H$ are bounded above by $K{K}n^{-1/2}$ in absolute
value, we obtain
\begin{equation}\label{boundexpectation}
\mu_{\nu({\bf y})} \leq K{K} \|{\bf y}\|\sqrt{2/k}\ .
\end{equation}
We use a concentration bound for vector-valued Rademacher sums
(tail inequality (1.9) in~\cite{LedouxT10}) to
notice that for any ${\bf y}$ and $t>0$,
\begin{equation}\label{tal} \Pr\left(| \nu({\bf y})-M_{\nu({\bf y})}|  > t\right) \leq 4\exp\left(-Ct^2/ \sigma_{\nu({\bf y})}^2 \right)\ ,\end{equation}
\hr{(constant 2 replaced by 4?)}\na{(You're right - fixed)}
where $M_{\nu({\bf y})}$ is a median of $\nu({\bf y})$ and, with $A= \sqrt{n/k} P_\Omega$,
\begin{eqnarray}
\sigma_{\nu({\bf y})} &=& \sup_{\|\beta\|^2_2+\|\gamma\|^2_2\leq 1}   \left (\sum_{j=1}^n\left |\sum_{i=1}^n \beta_i {\bf y}^*\xspace  {\bf x}_{ij} + \gamma_i {\bf y}^*\xspace {\bf x}_{ji}\right|^2\right)^{1/2}  \nonumber \\
&\leq& \sup_{\|\beta\|,\|\gamma\|\leq 1} \|  {\bf y}^*\xspace  A D_\beta H D_{Hx} \|_2 + \|  D_{{\bf y}^*\xspace  A}HD_{\gamma} Hx \|_2  \nonumber \\
&\leq& \sup_{\stackrel{\|\beta\|,\|\gamma\| \leq 1}{\|\beta'\|,\| \gamma'\| \leq 1}} {\bf y}^*\xspace  A D_\beta H D_{\beta'}Hx +  {\bf y}^*\xspace  AD_{ \gamma}HD_{\gamma'} Hx  \nonumber \\
&\leq& 2\sup_{\|\beta\|,\|\gamma\|\leq 1} {\bf y}^*\xspace  A D_\beta H D_{\gamma}Hx   \nonumber \\
&\leq &  2\sqrt{n/k} \|D_{{\bf y}^*\xspace  P_\Omega H}H D_{Hx}\|   \label{onebeforelast111}  \\
& \leq  & 2\|{\bf y}\|_2\cdot \|x\|_1 K{K}/\sqrt n \leq  2K{K}\sqrt{s/n}\ . \label{boundsigma}
\end{eqnarray}
Upper bounding the expression (\ref{onebeforelast111}) was done exactly as above when upper bounding $U$.
Using  (\ref{boundexpectation}) and the second part of Lemma~\ref{integrate}, we conclude that
\begin{equation}\label{boundMmu}
M_{\nu({\bf y})} \leq \mu_{\nu({\bf y})} + C \sigma_{\bf y} \leq K{K}\|{\bf y}\|\sqrt{2/k} + CK{K}\sqrt{s/n}\ .
\end{equation}
We will now bound $V$.
 To that end, we use
a general epsilon-net argument. Given a subset $T$ of a Euclidean space, we recall that
a set ${\cal N} \subset T$ is called  $\mu$-separated if $\|{\bf y} - {\bf y}'\|_2 > \mu$ for all ${\bf y}, {\bf y}' \in {\cal N}$, ${\bf y} \neq {\bf y}'$.
It is called maximally $\mu$-separated if no additional vector can be added to ${\cal N}$ in a $\mu$-separated position.

\begin{lem}\label{baraniuk}
Let $\gamma : {\mathbb C}^m \mapsto \mathbb R^+$ be a seminorm, and let ${\cal N}$ denote a maximal $\mu$-separated set of Euclidean
unit vectors ${\bf y} \in {\mathbb C}^m$ for some $\mu<1$.
Let
$$S = \sup_{{\bf y}\in {\cal N}} \gamma({\bf y})\;\;\; I = \inf_{{\bf y}\in{\cal N}}\gamma({\bf y})\ .$$
Then
\begin{eqnarray}\label{net1}
\sup_{\|{\bf y}\|=1} \gamma({\bf y}) &\leq& \frac 1{1-\mu} S \label{net1up}\\
\inf_{\|{\bf y}\|=1} \gamma({\bf y}) &\geq& I - \frac{\mu}{1-\mu} S \ . \label{net1down}
\end{eqnarray}
In particular, if $\kappa := \sup_{{\bf y}\in {\cal N}} |\gamma({\bf y})-1|$, then
\begin{equation}\label{net2}
\sup_{\|{\bf y}\|=1} |\gamma({\bf y})-1| \leq \frac {\kappa+\mu} {1-\mu}\ .
\end{equation}
\end{lem}
The proof of the bound \eqref{net1up} is contained in \cite{ve12}. The proof of \eqref{net1down} is similar and implicitly
contained in \cite{badadewa08}.

We use the lemma by constructing a maximal $\eta = 0.1$-separated set ${\cal N}$ of
$S_\Omega := \{ {\bf y} \in {\mathbb C}^n : \|{\bf y}\|_2 = 1, \operatorname{supp} {\bf y} \subset \Omega\}$.
Using a standard
volumetric argument (see e.g.\ \cite[Proposition 10.1]{ra10})
\begin{equation}\label{card}
 \card{{\cal N}} \leq (1+2/\eta)^{2k}
 = 21^{2k}\ .
\end{equation}
We also notice that $\nu(\cdot)$ is a seminorm \na{(indeed a seminorm)} for any fixed $\epsilon, \epsilon'$.
Using (\ref{net1up}),
$$ \sup_{\stackrel{{\bf y}: {\bf y}=P_\Omega {\bf y}}{ \|{\bf y}\|=1}} \nu({\bf y}) \leq \frac 1{1-0.1} \sup_{{\bf y}'\in{\cal N}} \nu({\bf y}') \  . $$
Taking expectations yields
\begin{equation}\label{boundsups} \mathbb E \sup_{\stackrel{{\bf y}: {\bf y}=P_\Omega {\bf y}}{ \|{\bf y}\|=1}} \nu({\bf y}) \leq 1.2 \mathbb E \sup_{{\bf y}'\in{\cal N}} \nu({\bf y}') \  .
\end{equation}
\noindent
The expectation on the right hand side can now be bounded, in light of (\ref{tal}) and using
Lemma~\ref{expmaximaeq} (with $\sigma_i' = 0$), as follows:
\begin{equation}\label{comparegaus2}
 \mathbb E  \sup_{{\bf y}'\in{\cal N}} \nu({\bf y}')  \leq \sup_{{\bf y}}M_{\nu({\bf y})} + \sup_{{\bf y}} \sigma_{\nu({\bf y})}\sqrt{\operatorname{log} \card{{\cal N}}}\ .
\end{equation}
Together with (\ref{boundMmu}),  (\ref{card}), and (\ref{boundsigma}), this implies
\begin{equation}\label{comparegaus}
 \mathbb E  \sup_{{\bf y}'\in{\cal N}} \nu({\bf y}')    \leq  CK{K}\left(\sqrt{1/k} + \sqrt{s/n} + \sqrt{\frac s n k}\right) \ .
\end{equation}
If we now assume that $k \leq \frac 1 2 \sqrt{\frac {n}{s}}$,
then we conclude from (\ref{boundsups}),
(\ref{comparegaus}) and  (\ref{Ubound}) that
$$ V=  \mathbb E \sup_{\stackrel{{\bf y}: {\bf y}=P_\Omega {\bf y}}{ \|{\bf y}\|=1}} \nu({\bf y}) \leq CK{K}/\sqrt k \quad \mbox{ and } \quad  U \leq 2K{K^2}/k\ .  $$
\hr{(Why the index 4 at $C_4$ here? Should we use a `generic' constant $C$ everywhere or maybe number through all constants?)}
\na{(A painful issue indeed - either the constant enumeration should be organized, or $C$ should be used everywhere - let's do that after the arxiv)}

\noindent
Plugging these upper bounds into (\ref{tal:chaos2}) we
conclude  that for all $0 < t\leq  1$
$$ \Pr(|\alpha(x) - M_{\alpha(x)}| \geq t) \leq 2\exp\left( - Ct^2 kK{/K^2}\right)\ $$
(because $\min\{t^2/V^2,t/U\} = Ct^2/V^2$ for $0 <t\leq 1$ and for the derived values of $U,V$).
But we also know, using Lemma~\ref{integrate}, that
$ | \sqrt{\mathbb E \alpha(x)^2} - M_{\alpha(x)} | \leq CK{K}/\sqrt k$.  Recalling that $\mathbb E \alpha(x)^2 = 1$ and
combining, we conclude that  for $t\leq 1$,
\begin{equation}\label{tail111} \Pr(|\alpha(x) - 1| \geq t) \leq C\exp\left\{ - C t^2kK{/K^2}\right\}\ .\end{equation}

Now we fix a support set $T \subset \{1,\hdots, N\}$ of size $s$ and consider the complex unit sphere restricted to $T$, i.e.,
$S_T = \{x \in {\mathbb C}^n: \|x\|_2 = 1, \operatorname{supp} x \in T\}$. Let ${\cal N}_T$ be a maximal $\eta$-separated set of $S_T$, which has
cardinality at most $(1+2/\eta)^{2s}$ by \eqref{card}.
By a union bound, we have
\begin{align}
\Pr(\max_{x \in {\cal N}_T} |\alpha(x)-1| \geq t) & \leq (1+2/\eta)^{2s} C e^{-C t^2 k} \notag\\
& = C \exp\left( -C t^2 k + 2s \operatorname{log}(1+2/\eta)\right)\ . \notag
\end{align}
It follows from Lemma \ref{baraniuk} that
\begin{align}
&\Pr(\max_{x \in U_s} |\alpha(x)-1| \geq (t+\eta)/(1-\eta))\notag\\
& = \Pr\left(\max_{\#T = s} \max_{ x \in S_T} |\alpha(x) - 1| \geq (t+\eta)/(1-\eta)\right)\notag\\
& \leq \sum_{\#T = s} \Pr\left(\max_{x \in {\cal N}_T} |\alpha(x) - 1| \geq t\right)
\leq \left( \begin{matrix} n \\ s \end{matrix}\right)  C \exp\left( -C t^2 k + 2s \operatorname{log}(1+2/\eta)\right)\notag\\
&\leq \exp\left(-C t^2 k + 2s \operatorname{log}(1+2/\eta) + s \operatorname{log}(en/s)\right) \ .\notag
\end{align}
Choosing $\eta = \min\{t,0.5)$ we conclude that $\max_{x \in U_s} |\alpha(x)-1| \leq 4t$ with probability at least $1-\varepsilon$ if
\[
k \geq Ct^2(s (\operatorname{log}(1+2/t) + \operatorname{log}(en/s)) + \operatorname{log}(C \varepsilon^{-1}) \ .
\]
Replacing $t$ by $\delta/4$ and noting that \eqref{thm1:rest:delta} implies $\operatorname{log}(1+8/\delta) \leq c \operatorname{log}(en/s)$ concludes
the proof.

\section{The regime $s={\cal O}(\sqrt{n}/\operatorname{log} n)$}
\label{sqrtn}

We now use the idea developed in the previous section to bootstrap an efficient RIP-optimal construction for  a larger regime.

Let $A$ be a fixed $k\times n$ matrix with pairwise orthogonal rows of Euclidean length $\sqrt{n/k}$ each.  Namely,
\begin{equation}\label{assumptionA}
A A^*\xspace  = \frac n k \operatorname{Id}_k\ .
\end{equation}
\hr{$A^* $ or $A^T $ throughout the paper?}
\na{It's a macro \texttt{$\backslash$tp} now}

The strategy will be to improve the RIP parameter $\delta_s(A)$ of $A$ by replacing $A$ with $\tilde A = A D_\epsilon H D_{\epsilon'} H$.
To analyze the (random) RIP parameter $\delta_s(\tilde A)$, fix an $s$-sparse unit vector $x \in {\mathbb C}^n$.  Now define the random variable
$$ \alpha(x) = \|\tilde A x\|_2 .$$

As before, we note that $\alpha(x)$ is the norm of a decoupled Rademacher chaos of degree $2$
in a $k$-dimensional Hilbert space,
which can be conveniently written as $\alpha(x) = \|\sum_{i=1}^n\sum_{j=1}^n \epsilon_i\epsilon'j {\bf x}_{ij}\|_2$, where
$$ {\bf x}_{ij} = AHP_{\{i\}} H P_{\{j\}} H x\ .$$

As in the previous discussion, we bound the invariants of interest $U$ and $V$ as defined in (\ref{defU}) and (\ref{defV}), respectively.  We start by bounding $U$.  By definition,
\begin{eqnarray}\label{boundU2}
U &\leq&  \sup_{\stackrel{{\bf y}, {\bf \alpha}, {\bf \beta}}{\|{\bf y}\|_2,\|{\bf \alpha}\|_2,\|{\bf \beta}\|_2\leq1}} {\bf y}^*\xspace  A D_{\bf \alpha} HD_{\bf \beta} H x\ .
\end{eqnarray}
Notice now that since $x$ is $s$-sparse by assumption, we have $\|x\|_1\leq \sqrt s$ and hence $\|Hx\|_\infty \leq K{K} \sqrt{s/n}$ so that
$$ \|D_{\bf \beta} Hx\|_2 \leq K{K} \sqrt{s/n} \quad \mbox{ and } \quad  \|D_{\bf \beta} Hx\|_1 \leq 1\ . $$
The right hand side inequality is due to Cauchy-Schwarz.
In turn, this implies
$$ \|HD_{\bf \beta} Hx\|_\infty \leq 1/\sqrt n \quad \mbox{ and } \quad  \|HD_{\bf \beta} Hx\|_2 \leq K{K}\sqrt{s/n}\ .$$
\noindent
Therefore,
\begin{equation}\label{L1L2}\|D_{\bf \alpha} HD_{\bf \beta} Hx\|_2 \leq 1/\sqrt n K{\leq K/\sqrt n}\quad \mbox{ and } \quad \|D_{\bf \alpha} H D_{\bf \beta} Hx\|_1 \leq K{K} \sqrt{s/n}\ .
\end{equation}
Again, the right hand side inequality is due to Cauchy-Schwarz.  We need the following simple lemma, see also \cite[Lemma 3.1]{plve11}.

\begin{lem}
Let $w\in {\mathbb C}^n$ be such that $\|w\|_1\leq \sqrt s \rho$ and $\|w\|\leq \rho$ for some
integer $s$ and number $\rho>0$.  Then there exist $N=\lceil n/s\rceil$ vectors $w^{(1)},\dots, w^{(N)}$ such that $w^{(i)}$ is $s$-sparse for each $i$, $w=\sum_{i=1}^N w^{(i)}$ and $\sum_{i=1}^N \|w^{(i)}\|_2 \leq 2\rho$.
\end{lem}
\begin{proof}
Assume wlog that the coordinates $w_1,\dots, w_n$ of $w$ are sorted
so that $|w_1| \geq |w_2| \geq \cdots \geq |w_n|$.
 For $i=1,\dots, N$, let  $$w^{(i)} = (\underbrace{0,\dots, 0}_{(i-1)s\mbox{ times}}, w_{(i-1)s+1},\dots, w_{is}, 0, \dots, 0)^*\xspace  \in {\mathbb C}^n$$ and $\alpha_i = \|w^{(i)}\|_\infty = |w_{(i-1)s+1}|$.
Clearly $w=\sum_{i=1}^N w^{(i)}$, and we have:
\begin{eqnarray}
\sum \|w^{(i)}\| &=& \|w^{(1)}\| + \sum_{i=2}^N \|w^{(i)}\|
\leq \rho + \sum_{i=2}^N \alpha_i \sqrt s\ . \label{eq:12norm}
\end{eqnarray}
But now notice that for all $i=1,\dots, N$, $\|w^{(i)}\|_1 \geq \alpha_{i+1}s$ (where we define $\alpha_{N+1}=0$), hence
we conclude, using the assumptions, that
$$ \sum_{i=2}^N \alpha_i s \leq \sum_{i=1}^N \|w^{(i)}\|_1  = \|w\|_1 \leq \sqrt s \rho .$$
\noindent
Therefore, $\sum_{i=1}^N \alpha_i \sqrt s \leq \rho$.
Together with (\ref{eq:12norm}), this implies the lemma.
\end{proof}
\begin{rem} Note that the technique of grouping together monotonically decreasing coordinates of a vector in blocks is
rather standard in compressive sensing,
see for example \cite{carota06-1}  or \cite{DBLP:journals/siamma/KrahmerW11}.
\end{rem}
From (\ref{L1L2}) we conclude that $D_{\bf \alpha} HD_{\bf \beta} H x$ can be decomposed as $\sum_{i=1}^N w^{(i)}$ as in the lemma
with $\rho = K{K}n^{-1/2}$.  Using the RIP assumption on $A$, for each $i=1,\dots, N$, $\|Aw^{(i)}\| \leq \|w^{(i)}\|(1+\delta)$.
By the lemma's premise, and using the triangle inequality, this implies
\begin{equation}\label{boundU3} U \leq 2(1+\delta)K{K}/\sqrt n\ .
\end{equation}

\noindent
Bounding $V$ is done as follows.  We define the process $\nu$ as
$$
\nu({\bf y}) =
\left(\|B_{{\bf y}} \epsilon\|_2^2 + \|B_{{\bf y}}^*\xspace  \epsilon'\|_2^2\right)^{1/2}\ ,
$$
where $(B_{{\bf y}})_{ij} = {\bf y}^*\xspace  {\bf x}_{ij}$,
over the set $\{{\bf y} \in \mathbb R^k: \|{\bf y}\|\leq 1\}$, so that $V = \mathbb E \sup_{{\bf y}} \nu({\bf y})$.
For any ${\bf y}$, $\nu({\bf y})$ is a Rademacher sum
in $k$-dimensional Hilbert space.  Thus, we can use \eqref{tal}
to conclude that for all ${\bf y}$,
$$
\Pr\left(\nu({\bf y}) > M_{\nu({\bf y})} + t\right) \leq 4\exp(-t^2/8\sigma_{\nu({\bf y})}^2)\ ,
$$
where $M_{\nu({\bf y})}$ is a median of $\nu({\bf y})$ and
\begin{eqnarray}
\sigma_{\nu({\bf y})} &=& \sup_{\|\beta\|^2+\|\gamma\|^2\leq 1}   \left (\sum_{j=1}^n\left |\sum_{i=1}^n \beta_i {\bf y}^*\xspace  {\bf x}_{ij} + \gamma_i {\bf y}^*\xspace {\bf x}_{ji}\right|^2\right)^{1/2}  \nonumber \\
&\leq& \sup_{\|\beta\|,\|\gamma\|\leq 1} \|  {\bf y}^*\xspace  AD_\beta H D_{Hx} \| + \|  D_{{\bf y}^*\xspace  A}HD_{\gamma} Hx \|  \nonumber \\
&\leq& \sup_{\stackrel{\|\beta\|,\|\gamma\| \leq 1}{\|\beta'\|,\| \gamma'\| \leq 1}} {\bf y}^*\xspace  AD_\beta H D_{\beta'}Hx +  {\bf y}^*\xspace  AD_{ \gamma}HD_{\gamma'} Hx  \nonumber \\
&\leq& 2\sup_{\|\beta\|,\|\gamma\|\leq 1} {\bf y}^*\xspace  AD_\beta H D_{\gamma}Hx \label{onebeforelast} \\
&\leq & 4(1+\delta)K{K}/\sqrt n\ . \label{boundsigma5}
\end{eqnarray}
For the last inequality, notice that (\ref{onebeforelast}) is bounded by twice the RHS of (\ref{boundU2}), and recall the derivation of
(\ref{boundU3}).
Using the first part of Lemma~\ref{integrate}, we conclude that for all ${\bf y}$ such that $\|{\bf y}\|=1$,
\begin{equation}\label{boundMnuy}
M_{\nu({\bf y})} \leq \mu_{\nu({\bf y})} + CK{K}(1+\delta)/\sqrt n\ ,
\end{equation}
where $\mu_{\nu({\bf y})}$ is the expectation of $\nu({\bf y})$.
Jensen's inequality yields
\begin{align}
\mu_{\nu({\bf y})} &\leq \|D_{H^*\xspace A^*\xspace {\bf y}} H D_{Hx}\|_F \leq \|H^*\xspace A^*\xspace {\bf y}\| \cdot \|Hx\|K{K}/\sqrt n
= \|A^*\xspace {\bf y} \| \cdot \|x\| K{K}/\sqrt n \notag\\
& \leq (\sqrt{n/k})K{K}/\sqrt n = K{K}k^{-1/2}\ .\notag
\end{align}
Again we notice that for any fixed $\epsilon, \epsilon'$, $\nu$ is a seminorm.
As before, let ${\cal N}$ denote a maximal $0.1$-separated set of Euclidean unit vectors in ${\mathbb C}^k$.
Hence by (\ref{net1up}) in  Lemma~\ref{baraniuk}, for any fixed $\epsilon, \epsilon'$,
$$ \sup_{\|{\bf y}\|=1} \nu({\bf y}) \leq 1.2 \sup_{{\bf y}\in {\cal N}} \nu({\bf y})\ .$$
Taking expectation on both sides and using Lemma~\ref{expmaximaeq} to bound the right hand side (recalling that the
cardinality of ${\cal N}$ is at most $21^{2k}$), we conclude
$$
V = \mathbb E \sup_{\|{\bf y}\| = 1} \nu({\bf y}) \leq \sup_{\|{\bf y}\|=1} M_{\nu({\bf y})} +  C\sqrt{k}\sup_{\|\nu({\bf y})\|=1} \sigma_{\nu({\bf y})} .$$
By (\ref{boundsigma5}) and by our bound on $M_{\nu({\bf y})}$,
we conclude
\begin{eqnarray}
V &\leq& K{K} {k^{-1/2}} + CK{K}(1+\delta)/\sqrt n + CK{K} \sqrt{k}(1+\delta)/\sqrt n \nonumber \\
&\leq&  K{K} ({k^{-1/2}}+ (1+\delta)C\sqrt {k/n})\ . \label{boundV3}
\end{eqnarray}
From (\ref{tal:chaos2}), (\ref{boundU3}), (\ref{boundV3}) we then conclude that for all $t>0$,
\begin{align}
& \Pr(|\alpha(x) - M_{\alpha(x)}| \geq t) \notag\\
& \leq 2\exp\left( - C \min\left\{ \frac{t^2}{(K{K}(1+\delta)\sqrt{k/n}+1/\sqrt k)^2}, \frac {t\sqrt n}{K{K}(1+\delta)}\right\}\right)\ .\label{tal3}
\end{align}
Using the first part of Lemma~\ref{integrate} and recalling that $\mathbb E \alpha(x)^2 = 1$ this implies that
\begin{equation}\label{median_var}
\left | M_{\alpha(x)} - 1\right | \leq CK{K}((1+\delta)\sqrt{ k/n}+1/\sqrt k)\ ,
\end{equation}

We now use the net-technique to pass to the supremum over all $s$-sparse unit vectors to provide
an estimate of the restricted isometry constant.
For each subset $T \subset \{1,\hdots,n\}$ of cardinality $s$ we consider a maximal $\mu$-separated
${\cal N}_T$ of the unit sphere $S_T$ of complex
unit length vectors with support $T$ where $\mu = 1/k$. By \eqref{card} and since $k < \sqrt{n/s}$ and $s \leq \sqrt{n}$,
the union ${\cal N} = \cup_{\#T = s} {\cal N}_T$ is bounded in size
by
\begin{equation}
\# {\cal N} \leq \left(\begin{matrix} N \\ s\end{matrix} \right) (1+2/\eta)^{2s} \leq (en/s)^{s} (1+2\sqrt{n/s})^{2s}
\leq \exp(C s \operatorname{log} n) \ . \label{card:bound}
\end{equation}
Using Lemma~\ref{expmaximaeq}, (\ref{tal3}) and (\ref{median_var}), we conclude that
\begin{eqnarray}
\mathbb E \sup_{x\in {\cal N}} |\alpha(x) -1 | &\leq& CK{K}((1+\delta)\sqrt{k/n}+1/\sqrt k)
+ CK{K}\sqrt{s\operatorname{log} n}(1+\delta)\sqrt{k/n}  \nonumber\\
& & + CK{K} (s\operatorname{log} n)(1+\delta)/\sqrt n \nonumber \\
& \leq & CK{K}\left((1+\delta)\left (\sqrt{\frac{ s k \operatorname{log} n}{ n}} + \frac {s\operatorname{log} n}{\sqrt n}\right ) + k^{-1/2}\right) \ .\label{boundalpha1_}
\end{eqnarray}
We will assume in what follows that
\begin{equation}\label{assume7}
s\operatorname{log} n \leq k\ ,
\end{equation}
so that (\ref{boundalpha1_}) takes the simpler form
\begin{eqnarray}
\mathbb E \sup_{x\in {\cal N}} |\alpha(x) -1 |
& \leq & CK{K}\left ((1+\delta)\sqrt \frac {sk\operatorname{log} n}{ n} +k^{-1/2}\right)\ .\ \label{boundalpha1}
\end{eqnarray}

Recalling that $\alpha$ is a seminorm and applying (\ref{net2}) in Lemma~\ref{baraniuk}
we pass to the set of all $s$-sparse Euclidean unit normed vectors,
\begin{align}
\mathbb E \sup_{\stackrel{\|x\|=1}{\|x\|_0\leq s}} |\alpha(x) -1 | & = \mathbb E \max_{\# T = s} \sup_{x \in S_T}  |\alpha(x) -1 |\notag\\
& \leq  CK{K}\left ((1+\delta)\sqrt \frac {sk\operatorname{log} n}{ n} +k^{-1/2}\right)\ .\ \label{boundalpha11}
\end{align}

\subsection{A Bootstrapping Argument}

Let $\delta'$ denote $\delta_s(\tilde A)$.
Assume henceforth that the parameters $s,k$ satisfy
\begin{equation}\label{sparsityassump}
\kappa := C K{K}\sqrt{ \frac {sk\operatorname{log} n}{ n}}  < 1/2\ .
\end{equation}
Clearly (\ref{boundalpha1}) is
 a bound on  $\mathbb E [\delta']$.
With the new notation, we get
$$ \mathbb E [1+\delta'] \leq (1+\delta) \kappa + 1 +  CK{K}k^{-1/2}\ .
$$
Denote $\tilde A$ by $A^{(1)}$ and $A$ by $A^{(0)}$.  Now consider inductively repeating the above process, obtaining  (for $i\geq 2$) $A^{(i)}$ from $A^{(i-1)}$ by $$A^{(i)} = A^{(i-1)}  D_{\epsilon_{(i)}} H D_{\epsilon'_{(i)}} H\ ,$$ where $\epsilon_{(i)}, \epsilon'_{(i)}$ are independent copies of $\epsilon, \epsilon'$.  Let $\delta^{(i)}$ denote $\delta_s(A^{(i)})$. By independence and
the principle of conditional expectation, we conclude that
$$ \mathbb E[1+\delta^{(i)}] \leq (1+\delta^{(0)})\kappa^i +  \frac{1 + CK{K}k^{-1/2}}{(1-\kappa)}\leq (1+\delta^{(0)})\kappa^i + 2(1 + CK{K}k^{-1/2})\ .$$

Assume in what follows that $k$ is large enough so that $CK{K}k^{-1/2} \leq 1$. Then last inequality conveniently implies $\mathbb E[1+\delta^{(i)}] \leq (1+\delta^{(0)})\kappa^i + 4$.
Recall by our definition of $A$  that $\delta=\delta^{(0)}$ can be no more than $\sqrt{n/k}$.
Let $r$ be taken  as
\begin{equation}\label{defi1}
r := \left  \lceil \frac{ -\operatorname{log}(2\sqrt{n/k})}{\operatorname{log} \kappa}\right \rceil
\end{equation}
so that $(1+\delta(0))\kappa^{r} \leq (1+\sqrt{n/k})\kappa^{r} \leq 2\sqrt{n/k}\kappa^{r} \leq 1$ and hence
$$ \mathbb E[1+\delta^{(r)}] \leq 5\ .$$

Using Markov inequality, this implies that with probability
at least, say,  $0.995$
\begin{equation}\label{event} 1+\delta^{(r)} \leq  1000\ .\end{equation}
From now on assume event (\ref{event}) holds.
Now for an $s$-sparse unit vector $x$, let $x^{(r+1)} := A^{(r+1)} x$.
The assume $k \leq \sqrt{n}$ is equivalent to $1/\sqrt k \geq \sqrt{k/n}$.  Using this, and substituting a constant for $(1+\delta)$, (\ref{tal3}) implies
\begin{align}
& \Pr(| \|x^{(r+1)}\|_2 - M_{\|x^{(r+1)}\|_2}| \geq t) \notag\\
 & \leq 2\exp\left( - C \min\left( \frac{t^2}{(K{K}k^{-1/2})^2 }, tK{K}\sqrt n\right)\right)\ . \label{tal4}
\end{align}
where $M_{\|x^{(r+1)}\|_2}$ is a median of $\|x^{(r+1)}\|_2$.

Once again we consider maximal $\mu$-separated sets ${\cal N}_T$ of $S_T$ with $\mu = 1/k$
for each $T \subset \{1,\hdots,N\}$ of size $s$
and form ${\cal N} = \cup_{\#T = s} {\cal N}_T$.
The cardinality of ${\cal N}$ is at most $\exp\{Cs\operatorname{log} n\}$, see \eqref{card:bound}.
We can now use a union bound over ${\cal N}$, to conclude that with probability at least $0.995$,
\begin{equation}\label{i1plus1_}
\max_{x\in {\cal N}} \left | \|x^{(r+1)}\|_2 - M_{\|x^{(r+1)}\|_2} \right | \leq  CK{K}\max\left(\sqrt{\frac{s \operatorname{log} n}{k}},\  \frac{s\operatorname{log} n}{\sqrt n}\right)\ .
\end{equation}
Using (\ref{median_var}), this implies
\begin{eqnarray}
\max_{x\in {\cal N}} \left | \|x^{(r+1)}\|_2 - 1 \right |  &\leq& CK{K}\left( \max\left(\sqrt{\frac{sk \operatorname{log} n}{n}},\  \frac{ s\operatorname{log} n}{\sqrt n}\right) +k^{-1/2}\right) \notag \\
&\leq &  CK{K}\left(\sqrt{\frac{sk \operatorname{log} n}{n}} + k^{-1/2}\right)\ ,\label{i1plus1}
\end{eqnarray}
where the last inequality used (\ref{assume7}). As before,
using (\ref{net2}) in Lemma~\ref{baraniuk}, allows us to pass to the set of all $s$-sparse vectors:
\begin{eqnarray}
\sup_{\stackrel{\|x\|=1}{\|x\|_0\leq s}} \left | \|x^{(r+1)}\|_2 - 1 \right |  &\leq&
  CK{K}\left(\sqrt{\frac{sk \operatorname{log} n}{n}} + k^{-1/2}\right)\ .\label{i1plus111}
\end{eqnarray}
\noindent
Recalling the assumption $k \leq \sqrt n$,  this implies
\begin{equation*}
\delta^{(r+1)}  \leq CK{K}\sqrt{\frac{s\operatorname{log} n} {k}}\ .
\end{equation*}
and the proof of Theorem~\ref{thm2} is concluded.

redpajama_set_name: RedPajamaArXiv