JOHN ATENCIO will hold his first limited edition 'Trunk Show' during late April and early May in select John Atencio Boutique locations. Each event will feature an extraordinary and limited selection of black diamond necklace styles – showcased along side established John Atencio Collections which complement the exotic and elegant evening look. As with all John Atencio diamonds, each of the black diamonds included in this Trunk Show are conflict free and responsibly sourced from mines that adhere to the Kimberley Process.
The tagline for the Trunk Show: The New Noir: Elegance Re-Faceted recognizes the unique multi-checker-cut faceting of the exotic black diamonds in each necklace style. This faceting creates a higher level of brilliance, luster and shine for a truly exotic and elegant look. A wide selection of necklace lengths and styles will be showcased.
The Black Diamond Trunk Show event begins during April Diamond month at John Atencio. Featured in five select locations for six days each, the Trunk Show will kick off in Cherry Creek and Ft Collins on April 25-30th. Each boutique will host an evening event Black Diamonds and Bellini's to enhance the shopping experience.
It's going to be a total frenzy! On Saturday June 3, the Castle Pines Chamber of Commerce will hold its third annual Food Truck Frenzy event at Coyote Ridge Park in Castle Pines from 5:00 – 8:00 p.m. Celebrating the beginning of summer, this amazing day promises to feature delicious food, wonderful live music, and lots of family-friendly fun!
Due to the record turnout last year, the Chamber is adding even more trucks to this year's Food Truck Frenzy! Ten food trucks will provide a myriad of food from pizza, crepes, grilled cheese, and street tacos to delicious ice cream and shave ice. People from all over the community can camp out in lounge chairs and blankets all afternoon with friends and family while soaking up the sun and dancing to 70s and 80s hits with the band Lookin Back. Little ones can enjoy hula hooping, climbing on the jungle gym, and many more fantastic activities. Stop by our new pet corner for special treats for your favorite furry friend.
Community events such as this are made possible due to the generous support of local business and community partners. The Castle Pines Chamber extends appreciation to Castle Rock Bank, Animal Care Center of Castle Pines, Castle Pines North Metropolitan District, the City of Castle Pines, The Castle Pines Connection, Castle Rock Adventist Hospital, CruiseOne, Douglas County Libraries, Douglas County Living, Keith Dodd/Realty One Group Premier, and Orangetheory Castle Rock. Special thanks to Rockyard Brewery providing beer and to Steve Evans of Dancing Dawgz food trucks for his tremendous help in organizing the food.
Hope to see you at Coyote Ridge on June 3 for this amazing community event!
Douglas County Libraries in Parker (20105 East Mainstreet) is seeking volunteers age 14 and up to help launch kids into a galaxy of fun and learning at its May 20 Astronaut School. Volunteers will help train aspiring astronauts through various science activities, games, and outer space crafts. No prior experience or special skills are required.
"Volunteering at Astronaut School is a playful way to make connections with the youth in our community," says Parker Branch Volunteer Coordinator Aimee Drury. "Play is a vital part of the learning process, and Douglas County Libraries infuses it into all aspects of this program for both participants and volunteers," she adds.
The Dumb Friends League Harmony Equine Center is opening its barn doors to the public for a one-of-a-kind event. Horse lovers of all ages are invited to the Harmony Equine Center Open House on Saturday, June 10. This free, family-friendly event takes place at the Harmony Equine Center, located at 5540 E. Highway 86, just west of Franktown, from 10 a.m. to 3 p.m. Visitors will have the opportunity to take guided tours of the beautiful 168-acre property and meet horses available for adoption. Guests can also enjoy local food trucks, games for kids and a chili cook-off.
During the open house, visitors will get a firsthand look at how the Harmony Equine Center trains formerly abused or neglected horses so they are ready for adoption. Harmony's staff of professional horse trainers will present demonstrations throughout the day covering a wide range of topics, including how to trailer-load a horse, selecting the right horse, working your horse in the round pen and horse ownership 101. Additionally, guests will be able to watch horses complete an obstacle course used for training and enrichment.


text:
Home/News/Became known the scheme of delivery of explosives to the Crimean bridge to commit a terrorist attack
Became known the scheme of delivery of explosives to the Crimean bridge to commit a terrorist attack
war correspondent Alexander Kots described a scheme for delivering explosives to the Crimean bridge to carry out a terrorist attack. According to him, two heavy vehicles participated in the operation.
"The cargo – 22 pallets of film, 9 rolls each (one pallet weighs about a ton) – arrived by sea in the Georgian port of Poti from Bulgaria. The customer was a citizen of Ukraine. Then it was loaded onto a truck with foreign numbers, the driver communicated with the customer via WhatsApp "From Poti the truck went to Armenia, where the cargo was cleared according to the rules of the Customs Union. Then the truck again entered Georgia, and from it – to the territory of the Russian Federation through the Upper Lars checkpoint. The end point was the city of Armavir, where the pallets were reloaded onto another truck, which went towards the Crimea," Kots reports.
Probably, the explosive was placed in rolls of film in such a way that the X-ray did not detect it, the military commander expresses his opinion. According to him, this was done in Bulgariawhich means that not only Ukraine, but also foreign intelligence services took part in the terrorist attack on the Crimean bridge.
The Investigative Committee of the Russian Federation is investigating the blowing up of the bridge. The driver of the first truck is already giving his testimony, he was most likely used blindly by the organizers of the attack, Kotz said.
Russia massively shelled seven regions, in hundreds of villages in the Kherson region
While Ukraine freezes, Zelensky's wife walked through Parisian boutiques and squandered $40,000
Beloved of heaven
Mapastepec, Chiapas, registers an earthquake of 4.6 magnitude
"She screamed so that she didn't hear herself": Guzeeva told why she was ashamed in front of her children

redpajama_set_name: RedPajamaCommonCrawl


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Caesar is an entertainer, performing his mind-blowing magic all across the country for Fortune 500 Companies and Exclusive Private Events. Today he currently resides in Houston TX.
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Change, a Four-in-Art Quilt | OccasionalPiece--Quilt!
Posted on May 1, 2014 by Elizabeth E.
Yet most people think of this when I say hillside letters. . .
. . . but to me, neither of those counts. A letter needs to be embedded on a hillside or a mountain to count as a landmark. So that was the genesis of the quilt.
Our landmark hillside letter is a C, an imitation of that first University of California-Berkeley letter, which was set onto a hillside about 1905, the granddaddy of all the other mountain monograms. This is a blurry image from Wikipedia of our local mountain (our letter isn't really yellow). I tried to photograph it, but couldn't get a good vantage point, so this will have to do.
What is the significance of these other letters, spelling out the word C-H-A-N-G-E?
As Longfellow observed, "all things must change." And I keep my mother's advice that "A change is as good as a rest" close to my heart, for that's a truth as well. But the C-for-Change linkage came to me one weary night, when I had to go and do one more pick-up and one more errand when teenaged children were still at home. It must have been during our University's Homecoming Week, for when I rounded the street corner at the base of the Box Springs mountain, I could see the "C" all lit up. I pulled over and gazed at the glowing letter with that tired-behind-the-back-of-the-eyes fatigue, wishing that that I could go home and be home, like I could when the children were little and weren't off at some activity that required me to be out and about picking them up. I thought back to the "easy days" of tucking them in after a story and prayers and a drink, and about how wonderful those times were. Why did things always have to change?
But I realized that change is the law of the universe, and instead of being at war with that constant mutability towards "something new, something strange" I should just accept it. Change and I are now uneasy companions. I know it won't always be like that, for experience has taught me that change can come in steep cliffside drop-offs and hair-raising turns on a winding road. But for now, I'll be content gazing at my quilt where it hangs in the corner of my kitchen.
And come back for the next post, with instructions on how you can make your own little art quilt.
Elizabeth, what a delightful story and a wonderful little quilt! At first, I thought "oh my gosh – she got the theme wrong!", but trust you to have such a unique spin on it! And nope, back East I can't say we have mountains (or hills) with letters on them!
Great post! All so true and the little art quilt is lovely too.
Lovely against the yellow wall.
I remember someone telling me, "If we don't change, we die." That's a bit extreme, but maybe "wither" would be a good word to replace "die." Nice post and fun quilt.
So well said Elizabeth . . . change is not something that comes easy to me but I love how you have put the notion into words. A very appropriate tie in to your quilt as well because changes in our lives really are emotional landmarks that signify important aspects of the time. Chnages can mark our lives in profound ways. Well done as always.
I didn't know that Cal's Big C was the grandaddy! When I was there eons ago, repainting it was an annual ritual, with buckets of paint, and probably buckets of beer, carried up the hill. (I stayed home – it was a guy's event.) Thanks for bringing back memories. I love your quilt.
So many layers of meaning, I'm so impressed. Here in WI I don't think we have any hills with letters, but it does seem to me that there are a couple in Iowa that I would see when visiting relatives, but they were laid out with rocks. Loved the reference to the Cars movie, too.
Change is hard, until it becomes the new normal, then we wonder why we complained so much in the first place!
Great quilt as always, and definitely a unique one. Love it.
Fascinating interpretation, as always. In the town where both sets of grandparents lived, about thirty miles from here (Reedley) is Smith Mountain, with a big S on it.
Elizabeth, your art quilt is terrific! I love your post and how you incorporated the history of the mountain letters with your landmark theme as well as various thoughts on change. I am fascinated with your research of those letters. Growing up in the south and on the gulf coast, I saw no hills much less mountains. It wasn't until a few years ago (and quite the grown up) that I saw the M on the mountains in Bozeman, Montana. When my sister flew out to visit us, I jokingly told her we had put the M up there to welcome here as her last name begins with an M! Thanks for sharing.
Elizabeth, it is alway fun to read your posts and this one is no exception. I love how you have researched your inspiration so well and shared it with us all. Your 'Change" quilt reminds me of the lyrics to a Mercy Me song that we exercise to in Jazzercise and I also heart on the radio. I also love the reminder of Mountain Monograms because that is something that I grew up with in Colorado as well!
Love where you went with this theme. I guess I always assumed people put letters on their hills too. I've never lived around mountains, but hills in Canyon Lake, TX have a few letters/symbols. I love that you spelled out change while showing the landscape changing through your fabric choices, like seasons etc.. I like change…most of the time 🙂 Great job!
I love your quilt and the story behind it. Change is interesting, and I love mountains. I am pretty sure that having letters on the mountains is illegal in Canada, however. How fun to see one that lit up every night. This is a great take on the theme, an unexpected turn, which I love.
You do such a wonderful job with your quilts and explaining your interpretation of the themes. This quilt is great and it looks really nice on the yellow wall.
Elizabeth…your story makes this little gem sooo special! Thank you for sharing.
A friend just introduced me to your site. I had no idea there are mountains with letters. I grew up in what I consider the East (WV). There are mountains in the East, too. I grew up in the Appalachian Mountains. I've been to the Great Smoky Mountains. The Allegheny Mountains are close to the Appalachians. That's just the ones close to where I grew up. There are a plethora of mountains there and in my birth home area most of the state parks are in mountain ranges. Thank you for introducing us to the world of letters. Thanks for including your creative process. I took a grad class just a few years ago in creativity. I wish I had not been traveling with my work and had been quilting so I could have used that in my project. I am NOT much of a crocheter. I much prefer knitting. However, my daughter was in a sorority at OSU (Oklahoma) and I was taking the class through OSU – based in Tulsa. So, she and I drew out a design using some of the symbols of her sorority. I took that blanket with me on trips. I didn't think I would get it done, so I cross stitched (yes, another craft I adore) a design by someone else of for her sorority and made it into a pillow with double ruffles. She gave it away when she left the sosority. She is not attached to material things as I am. She thought that ALL her t-shirts from sorority were gone. Nope, I have two big boxes. She lives in NM and was home about five years ago and found them in my "quilt" closet. So, now she wants a sorority t-shirt quilt. I found another of the cross-stitch pieces that I did of the pillow that is complete and I plan to use it on the back and maybe some quilt blocks. She wants only t-shirts on the front. So, I have a new project.


Celebrate the beginning of summer with the Capitol Classic!
The Saint Alphonsus Capitol Classic Race is open to kids ages 6 to 14, of ALL abilities. The one-mile course starts at the Historic Boise Train Depot and ends at the State Capitol with a Finish Fair in Capitol Park. Everyone is a winner and receives an Olympic-style medal, T-shirt and complimentary refreshments. Proceeds from the Classic will benefit the YMCA's Safe Routes To School is a program designed to encourage students to walk and bike to school to promote a healthy lifestyle, reduce traffic congestion, improve air quality, and enhance quality of life in our communities.
We are excited to announce that the Treasure Valley Family YMCA and Saint Alphonsus have teamed up to put on this beloved event. Our organizations' missions are closely aligned and we are eager to work together to improve the health of our community – especially our youth!
Packet pick up will take place at Big Al's on Thu, May 30 between 1-8pm. Racers are required to pick up their race information in advance as there is no packet pick up on race day. Make plans to come by and stay! Big Al's offers discounted bowling and shoe rental for Capitol Classic participants and their families. Celebrate the beginning of summer with Big Al's and the Capitol Classic!
Interested in volunteering to be a part of the fun?
Learn about how you can help. Volunteer Application coming soon.
The Saint Alphonsus Capitol Classic Race was started by Ron and Betsy Twilegar in 1983 to encourage young people to become interested in physical fitness, especially running. At that first race, held on September 2, 1983, 1,000 children ran from the State Capitol Building to the Union Pacific Depot. Through the efforts of Dr. Michael and Kirsten Coughlin, Saint Alphonsus became involved with supporting the race in 1987. The race date was changed to the end of the school year and has continued as a tradition for many Treasure Valley families.


The operatic aspirations of a box office employee come to life when the star of the evening's performance suffers an unfortunate accident. Betty convinces the dashing General Manager to allow her save the show, but her moment is ruined by a handsome co-worker who charms the GM with an alternate solution involving a "little bluegrass Eurydice and some country Orfeo". This leads to an important operatic staple – murder, carried out in this case with the sharp end of a pencil.


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Everything Is Enlightenment
by Joan Sutherland| September 27, 2022
Enlightenment is everywhere we look, says Joan Sutherland — we can choose to notice it, but at the same time, we can also trust that it will find us, wherever we are.
I Vow Not to Burn Out
by Mushim Patricia Ikeda| August 11, 2022
Mushim Patricia Ikeda says it's not enough to help others. You have to take care of yourself too.
How to Be a Bodhisattva
by Thich Nhat Hanh, Chögyam Trungpa Rinpoche, Judy Lief, bell hooks, Christina Feldman and The Karmapa| May 12, 2022
It may seem like an unattainable ideal, but you can start right now as a bodhisattva-in-training. All you need is the aspiration to put others first.
Turn Your Thinking Upside Down
by Pema Chödrön| September 28, 2021
We base our lives on seeking happiness and avoiding suffering, but the best thing we can do for ourselves is to turn this whole way of thinking upside down.
You're Ready Enough
by Pema Khandro Rinpoche| April 30, 2021
Wherever you find yourself, says Pema Khandro, that's the starting point of the bodhisattva path—all you need to do is take that first step.
Patience Isn't Passive
by Constance Kassor| November 27, 2020
Constance Kassor explains why patience isn't a passive tolerance of harm. Instead, patience requires a recognition of the deep interconnectedness of the world and an active engagement with it.
The Challenges of Caregiving
by Tynette Deveaux| June 29, 2020
Tynette Deveaux shares the difficulties of caregiving and the truth of suffering.
We Cry Out for Justice
by Jan Willis| May 29, 2020
There will only be justice in America, says Jan Willis, when we see all people as our equals. She offers an ancient Buddhist meditation to help us do that.
Making the Warrior Commitment
by Pema Chödrön| May 26, 2020
Pema Chödrön shows us how we can let go of self-centered worries and become a bodhisattva-warrior. It's the greatest happiness of all.
He's Playing His Cards Right
by Hal Atwood| June 27, 2019
Buddhist translator Scott Wellenbach won more than $650,000 playing poker. He's giving it all away to charity.


text:
\section{Introduction}

The deep learning revolution has led many to apply machine learning methods to classical problems in all fields including statistics. Examples range from estimation \cite{dong2015image, ongie2020deep, gabrielli2017introducing, dua2011artificial,dreifuerst2021signalnet,diskin2021learning} to detection \cite{samuel2019learning,girard2021deep,brighente2019machine,de2017approximating,ziemann2018machine,theiler2021bayesian}. Deep learning is a promising approach for developing high accuracy and low complexity alternatives when classical solutions are intractable. To facilitate this switch, we must ensure that the learned solutions are accurate but also satisfy other classical requirements. In this paper, we focus on learning detectors for composite hypothesis testing. We claim that current solutions deliver on their promises but lack the Constant False Alarm Rate (CFAR) requirement which is critical in many applications. To close this gap, we provide a framework for learning accurate CFAR detectors.

Detection theory begins with simple hypothesis testing where a detector must decide between two fully specified distributions. The classical solution is the Likelihood Ratio Test (LRT) which is optimal in terms of maximizing the detection probability subject to a false alarm constraint. Composite hypothesis testing is a more challenging setting where the hypotheses involve unknown deterministic parameters. A CFAR detector is invariant to these parameters and has constant false alarm probabilities. This allows the user to set the thresholds a priori. A popular approach is the Generalized Likelihood Ratio Test (GLRT) which can be interpreted as estimating the unknown parameters and then plugging them into a standard LRT. GLRT performs well when the number of unknowns is relatively small and is asymptotically CFAR. In finite sample settings, however, it is generally sub-optimal. Additionally, it may be computationally expensive. Some techniques were developed for designing CFAR detectors for specific families of distributions \cite{conte2003cfar,kokaly2017usgs,coluccia2022design}.

Data-driven classifiers are the machine learning counterpart to model-based detectors. In simple settings with no unknown deterministic parameters, it is well known that the optimal Bayes classifier converges to the LRT with a specific false alarm rate \cite[p. 78]{kay1998fundamentals}. More advanced classifiers can also maximize the cumulative detection rate over a wide range of false alarms, also known as the (partial) area under the curve (AUC) \cite{herschtal2004optimising, brefeld2005auc, narasimhan2013structural}. Large deviation analysis is available in  \cite{braca2022statistical}.
Consequently, there is a growing body of works on using machine learning for target detection. In the context of hyperspectral imagery, \cite{ziemann2018machine} introduced the use of
support vector machines (SVM) and deep neural networks were proposed in \cite{girard2021deep}.
  In the context of radar detection, SVMs were considered in \cite{de2017approximating}. Specific CFAR radar detectors were developed by relying on CFAR features \cite{lin2019dl,akhtar2018neural,akhtar2021training}.

The main contribution of this paper is a framework for learning CFAR detectors, denoted as CFARnet. The framework is general purpose and can be applied to arbitrary composite hypothesis testing problems. CFARnet is based on adding a penalty function that decreases the distances between the distributions under different parameter values. To optimize CFARnet, we rely on empirical and differentiable distances that have recently become popular in unsupervised deep learning \cite{goodfellow2014generative,li2015generative,gretton2012kernel}. Theoretically, we prove that under asymptotic linear and Gaussian settings, GLRT is indeed a minimizer of the Bayes risk subject to a CFAR constraint. Practically, CFARnet is a differentiable approximation to this minimizer that can also be applied in more general settings.

Our experiments show that CFARnet is approximately CFAR while paying a small price in terms of accuracy. First, we show this in the context of detecting a signal with unknown amplitude in non-Gaussian noise. Next, we turn to a more challenging adaptive detection problem with unknown Gaussian noise covariance but access to secondary data of noise-only samples, which can be used for estimation of the covariance of the noise. \cite{kelly1986adaptive,robey1992cfar,conte1996adaptive,manolakis2001hyperspectral}. Interestingly, we show that the popular regularizing trick, also known as shrinkage or diagonal loading, improves detection accuracy but leads to non-CFAR detectors. Using CFARnet, we learn a network which is both accurate and CFAR. Finally, we demonstrate the advantages of CFARnet in target detection of a plastic material in real hyperspectral imagery. Here too, CFARnet finds a tradeoff between accuracy and CFAR. Its advantages are evident when we examine the detection rates under a strict worst-case false alarm across the full image.

For completeness, we note that the CFAR notion is closely related to the topics of ``fairness'' and ``out of distribution (OOD)'' generalization which have recently attracted considerable attention in the  machine learning literature, e.g., \cite{arjovsky2019invariant,wald2021calibration}.
CFAR can be interpreted as a fairness property with respect to the unknown deterministic parameters. The closest work is \cite{romano2020achieving} which also enforces ``equalized odds'' using a distance between distributions. A main difference is that CFAR is a one-sided fairness property and requires equal rates only in the null hypothesis.
Algorithmically, \cite{romano2020achieving} compares the high dimensional joint distribution of the predictions and the unknown parameters, whereas we only consider the scalar distribution of the predictions. This makes our method significantly cheaper in terms of computational complexity. Indeed, we rely on a simple kernel based distance and do not require sophisticated adversarial networks.

\section{Problem formulation}
We consider a binary hypothesis test. Let $\boldsymbol{x}$ be an observed random vector whose distribution $p(\boldsymbol{x};\mathbf{z})$ depends on an unknown deterministic parameter $\mathbf{z}$. The value of $\mathbf{z}$ defines two possible hypotheses
\begin{align}\label{comp_testing}
    &y=0:\quad \mathbf{z}\in{\mathcal{Z}}_0\nonumber\\
    &y=1:\quad \mathbf{z}\in{\mathcal{Z}}_1.
\end{align}
A classical example is the following setting:
\begin{example} Target detection when both the target amplitude and the noise scaling are unknown:
\begin{align}\label{DC}
    \boldsymbol{x}=A\mathbf{1}+\sigma\boldsymbol{n}
\end{align}
where $\mathbf{1}$ is a target vector of ones, $\boldsymbol{n}$ is a random vector with i.i.d. noise variables, $\mathbf{z}=[A,\sigma]$ are deterministic unknown parameters, and
\begin{align}\label{params}
    &{\mathcal{Z}}_0=\left\{\mathbf{z}\;:\;A=0,\; 0.5\leq \sigma\leq 1\right\}\nonumber\\
    &{\mathcal{Z}}_1=\left\{\mathbf{z}\;:\;A\neq 0,\;-1\leq A\leq 1,\; 0.5\leq \sigma\leq 1\right\}
\end{align}
Here, $A$ acts as a discriminative parameter that varies between the two hypotheses, whereas the noise variance $\sigma^2$ is an unknown nuisance parameter which is identical in both.
\end{example}

The goal is to design a detector $\hat y(\boldsymbol{x})\in\{0,1\}$ as a function of $\boldsymbol{x}$ that will identify the true hypothesis $y\in\{0,1\}$.
Performance is measured in terms of probability of correct detection, also known as True Positive Rate (TPR):
\begin{align}\label{PD_FA1}
   {\rm Pr}_{\rm TPR}(\mathbf{z}) =  \; &{\rm Pr}(\hat y(\boldsymbol{x}) =1 ;y=1)
\end{align}
and probability of false alarm, also known as False Positive Rate (FPR):
\begin{align}\label{PD_FA2}
   {\rm Pr}_{\rm FPR}(\mathbf{z}) =  {\rm Pr}(\hat y(\boldsymbol{x}) =1 ;y=0).
\end{align}
In practice, the user typically provides a false alarm constraint ${\rm Pr}_{\rm FPR}\leq \alpha$ that must be satisfied and the goal is to maximize ${\rm Pr}_{\rm TPR}$.

It is standard to consider decision functions of the form
\begin{equation}\label{detector}
    \hat{y}\left(\boldsymbol{x}\right)=\mathbf{1}_{T(\boldsymbol{x})\geq\gamma}=\begin{cases}
0 & T\left(\boldsymbol{x}\right)<\gamma\\
1 & T\left(\boldsymbol{x}\right)\geq\gamma
\end{cases}
\end{equation}
where $T(\boldsymbol{x})$ is denoted as the {\textbf{detector}} function and $\gamma$ is a {\textbf{threshold}} value. This structure allows users to tune the FPR by adjusting the threshold.  Performance is usually visualized using the Receiver Operating Characteristic (ROC) which plots the TPR as a function of the FPR. In signal processing applications, users are often interested in a region of very low FPRs, e.g., $10^{-1}-10^{-3}$ and the goal is to maximize the TPR probabilities in this area.
A main challenge in detection theory is the unknown nuisance  parameters under the null hypothesis $y=0$, e.g., the unknown noise variance $\sigma^2$ in Example 1. The false alarm probability (FPR) is generally a function of these parameters and cannot be controlled without their knowledge. Therefore, it is often preferable to restrict the attention to CFAR detectors.
\begin{definition}
A detector $T(\boldsymbol{x})$ is CFAR if its FPR ${\rm Pr}\left(T(x)>\gamma|\mathbf{z}\in\mathcal{Z}_0\right)$ is invariant
to the value of $\mathbf{z}\in{\mathcal{Z}}_0$, for any threshold $\gamma$.
\end{definition}
It is straightforward to see that the above definition is equivalent to invariance of the distribution of $T(\boldsymbol{x})$ to all $\mathbf{z}\in{\mathcal{Z}}_0$.
For example, in Example 1 above, $T(\boldsymbol{x})$ is a CFAR test if its distribution is invariant to the value of $\sigma^2$.

As we will review below, many classical detectors are CFAR or asymptotically CFAR. With the growing trend of switching to machine learning, the goal of this paper is to introduce a framework for learning CFAR detectors, that can be used when classical detectors are intractable or far from optimal.

\section{Existing detectors}

\subsection{Classical detectors}
Traditionally, detectors were developed using likelihood ratios. In the simple case, all the parameters of the hypotheses are known. For example, this is the case of target detection in (\ref{DC}) where , ${\mathcal{Z}}_0=\{\mathbf{z}:A=0,\sigma=1\}$ and ${\mathcal{Z}}_1=\{A=1,\sigma=1\}$. In this case, hypothesis testing has an optimal solution known as the Likelihood Ratio Test (LRT) due to Neyman-Pearson lemma \cite[p. 65]{kay1998fundamentals}:
\begin{align}\label{LRT}
    T_{\rm LRT}(\boldsymbol{x}) &=2\log\frac{p(\boldsymbol{x};\mathbf{z}=\mathbf{z}_1)}{p(\boldsymbol{x};\mathbf{z}=\mathbf{z}_0)}
\end{align}
where the threshold $\gamma$ is chosen to satisfy the false alarm (FPR) constraint.

The more realistic scenario is composite hypotheses testing where one or both of the hypotheses allow multiple possible values and there is no solution which is optimal for all of them. A popular heuristic is the Generalized Likelihood Ratio Test (GLRT) that estimates the unknowns using the Maximum Likelihood (ML) technique and plugs them into the LRT detector \cite[p. 200]{kay1998fundamentals}:
\begin{align}\label{GLRT}
    T_{\rm GLRT}(\boldsymbol{x}) &=  2\log\frac{\max_{\mathbf{z}\in{\mathcal{Z}}_1}p(\boldsymbol{x};\mathbf{z})}{\max_{\mathbf{z}\in{\mathcal{Z}}_0}p(\boldsymbol{x};\mathbf{z})}
\end{align}
Setting the threshold to ensure a fixed ${\rm Pr}_{\rm{FPR}}$ is not trivial. Fortunately, under regularity conditions, GLRT is asymptotically CFAR and its threshold can be set for all values of the unknown parameters simultaneously.

GLRT is probably the most popular solution to composite hypothesis testing. It gives a simple recipe that performs well under asymptotic conditions. Its main downsides are that it is sensitive to deviations from its theoretical model, it is generally sub-optimal under finite sample settings and that it may be computationally expensive. Both the numerator and denominator of the GLRT involve optimization problems that may be large scale, non-linear and non-convex. Therefore, there is an ongoing search for robust and low cost alternatives.

\subsection{Learned detectors}\label{ml4lrt}

An alternative approach to designing detectors is based on machine learning. The starting point to any data-driven learning is a training set. Hypothesis testing relies on a probabilistic model $p(\boldsymbol{x};\mathbf{z})$ and we need to use this model in order to generate a synthetic dataset. Hybrid settings involving a mixture of real and artificial samples are also common. For example, it is standard to plant synthetic targets on real noise or clutter samples  \cite{ziemann2018machine}.

A main challenge in generating data is that $y\in\{0,1\}$ and $\mathbf{z}$ are not random variables but deterministic parameters without any prior distribution. A natural heuristic is to assume uniform fake priors, e.g., choose half of labels as $y=0$ and half as $y=1$ and assume that $\mathbf{z}$ is uniformly distributed on its corresponding domains. For each $y_i$ and $\mathbf{z}_i$, we then generate a measurement $\boldsymbol{x}_i$ according to the true $p\left(\boldsymbol{x};{\mathbf{z}_i}\right)$ and obtain a synthetic dataset
\begin{equation}
    {\mathcal{D}}_N=\{\boldsymbol{x}_i,\mathbf{z}_i,y_i\}_{i=1}^N.
\end{equation}

Next, a class of possible detectors ${\mathcal{T}}$ is chosen in order to tradeoff expressive power with computational complexity in test time. The class is usually a fixed differentiable neural network architecture. In our context, it also makes sense to reuse existing ingredients from classical detector as non-linear features or internal sub-blocks \cite{akhtar2021training, samuel2019learning}.

Finally, the learned detector is defined as the minimizer of an empirical loss function
\begin{equation}\label{classifcation}
   \min_{\hat{{T}}\in {\mathcal{T}}} \frac 1N \sum_{i=1}^N L(\hat T(\boldsymbol{x}_i),y_i).
\end{equation}
where $L(\cdot;\cdot)$ is a classification loss function. Ideally, we would like to minimize the zero-one loss which corresponds to the average probability of error. Practically, for efficient optimization, a smooth and convex surrogate loss, as the hinge or cross entropy functions, is preferable. The overall procedure for learning a detector is summarized in Algorithm 1.

\begin{algorithm}
\caption{NET detector for $p(\boldsymbol{x};\mathbf{z})$}\label{alg:BCE}
\begin{itemize}
    \item Choose   ${\rm Pr}^{\rm{fake}}(y)$.
    \item Choose  $p^{\rm{fake}}(\mathbf{z};y)$.
    \item For each $i=1,\cdots,N$:

    \hspace{4mm}Generate $y_i$.

    \hspace{4mm}Generate $\mathbf{z}_i$ given $y_i$.

    \hspace{4mm}Generate $\boldsymbol{x}_i$ given $\mathbf{z}_i$.
    \item Solve

    \hspace{4mm}$\min_{\hat{{T}}\in {\mathcal{T}}} \frac 1N \sum_{i=1}^N L(\hat T(\boldsymbol{x}_i),y_i)$.
\end{itemize}
\end{algorithm}

Learned detectors have strong theoretical guarantees in simple testing problems where $y$ uniquely defines $\mathbf{z}$. With a sufficiently expressive class of detectors and a large enough training set, minimizing the zero-one loss leads to the Bayes optimal detector. It is identical to the LRT with the threshold
\begin{equation}
    \gamma = 2 {\rm log}\left(\frac{{\rm Pr}^{\rm fake}(y=0)}{{\rm Pr}^{\rm fake}(y=1)}\right).
\end{equation}
Minimizers of Bayes risk consistent surrogates, as the hinge loss,  also asymptotically converge to this LRT \cite{bartlett2006convexity}. Thus, any simple LRT can be approximated by tuning ${\rm Pr}^{\rm fake}(y)$ to achieve a desired FPR.

There is limited theory on the performance of learned detectors in composite hypothesis testing.
Numerical experiments suggest that learned detectors usually provide similar accuracy as their corresponding (G)LRTs.
On the negative side, our experiments show that learned detectors are not CFAR and result in significantly different false alarm rates for different values of $\mathbf{z}\in{\mathcal{Z}}_0$. To close this gap, in the next section we propose a framework for learning CFAR detectors.

\section{Learning CFAR detectors}

In this section, we introduce CFARnet, a framework for designing CFAR detectors using machine learning. CFARnet introduces two modifications to Algorithm 1. First, we augment the classification loss with a penalty function that ensures similar distributions of the detector $\hat T(\boldsymbol{x})$ for all values of $\mathbf{z}\in {\mathcal{H}}_0$. Second, in order to compare such distributions empirically, we need to generate multiple $\{\boldsymbol{x}_{ij}\}_{j=1}^M$ for each $\mathbf{z}_i$.

The main idea is adding a penalty to the objective function that promotes a CFAR detector. For this purpose, we need to measure the distance between different distributions.
\begin{definition}
Let $X\sim p(X)$ and $Y\sim p(Y)$ be two random variables. A statistical distance $d(X;Y)$ is a function
that satisfies $d(X;Y)\geq 0$ with equality if and only if $p(X)=p(Y)$. In particular, an integral probability metric $d(X;Y)$ can be empirically estimated using a set of data realizations $\hat d(\{X_j\}_{j=1}^M,\{Y_j\}_{j=1}^M)$.
\end{definition}
Given a statistical distance $d(\cdot;\cdot)$, the penalty is defined as a sum of distances between the distributions of $\hat{T}$ under different values of $\mathbf{z}$:
\begin{align}\label{const2}
    R(\hat{T})=\sum_{\mathbf{z}, \mathbf{z}'\in {\mathcal{Z}}_0} d\left(\hat{T}(\boldsymbol{x}); \hat{T}(\boldsymbol{x}')\right)
\end{align}
where
\begin{align}
    &\boldsymbol{x}\sim p(\boldsymbol{x};\mathbf{z})\nonumber\\
    &\boldsymbol{x}'\sim p(\boldsymbol{x};\mathbf{z}').
\end{align}
Clearly, any CFAR test must satisfy $R(\hat{T})=0$.

Practically, to minimize (\ref{const2}), we use empirical estimates of the distances where each distribution is represented using a small dataset. For each $\mathbf{z}_i$, we synthetically generate multiple observations $\boldsymbol{x}_{ij}$ for $j=1,\cdots,M$. Similarly, for each $\mathbf{z}_{i'}$ we compute multiple $\boldsymbol{x}_{i'j}$. We then plug these into the empirical distances:
\begin{align}\label{const22}
    \hat R(\hat T)=\sum_{i,i'} \hat d\left(\{\hat{T}(\boldsymbol{x}_{ij}\}_{j=1}^M); \{\hat{T}(\boldsymbol{x}_{i'j})\}_{j=1}^M\right)
\end{align}
Our implementation of CFARnet uses the Maximum Mean Discrepancy (MMD) distance \cite{gretton2012kernel} as detailed in Appendix \ref{app_dist}. We also use a hyper-parameter $\alpha>0$ that trades off the importance of the classification accuracy versus the CFAR penalty. The overall CFARnet procedure is summarized in Algorithm 2.

\begin{algorithm}
\caption{CFARnet detector for $p(\boldsymbol{x};\mathbf{z})$}\label{alg:cfar}
\begin{itemize}
    \item Choose   ${\rm Pr}^{\rm{fake}}(y)$.
    \item Choose  $p^{\rm{fake}}(\mathbf{z};y)$.
    \item For each $i=1,\cdots,N$:

    \hspace{4mm}Generate $y_i$.

    \hspace{4mm}Generate $\mathbf{z}_i$ given $y_i$.

    \hspace{4mm}For $j=1,\cdots,M$:

    \hspace{8mm} Generate $\boldsymbol{x}_{ij}$ given $\mathbf{z}_i$.
    \item Solve

    \hspace{4mm}$\min_{\hat{{T}}\in {\mathcal{T}}} \frac 1N \sum_{i=1}^N L(\hat T(\boldsymbol{x}_i),y_i)+\alpha \hat R(\hat T)$.
\end{itemize}
\end{algorithm}

\section{Relation to GLRT}
A natural question is the relation between the learned CFARnet detector and the classical GLRT. To answer it we consider a theoretical version of CFARnet which minimizes the expected Bayesian 0-1 loss subject to an exact CFAR constraint:
\begin{align}\label{CFAR optimization}
    {\rm BayesCFAR:}\; \left\{\begin{array}{ll}
        \min_{\hat{T},\gamma} & {\rm Pr}(\mathbf{1}_{T\geq\gamma}\neq y) \\
        {\rm s.t.} & \hat{T} \; {\rm is\;CFAR}
    \end{array}\right.
\end{align}
where the fake priors enter the optimization through the error probability. CFARnet with large $\alpha$ is a finite sample differentiable approximation of BayesCFAR. The main result of this section is that under favorable conditions, GLRT is a minimizer of BayesCFAR. Thus, CFARnet can be interpreted as a deep learning approximation to GLRT, which can be learned even when the latter is intractable (due to unknown models or high computational complexity). We conjecture that this relation holds for most settings in which GLRT attains its asymptotic performance. In what follows, we rigorously prove this claim for the general Gaussian linear problem.

\begin{definition}
The linear model with Gaussian noise model is defined as
\begin{equation}\label{linear model}
    \boldsymbol{x} = \H\mathbf{z}_r + \boldsymbol{n}
\end{equation}
where $\boldsymbol{x}\in\mathbb{R}^n$ is the observed vector, $\H\in\mathbb{R}^{n\times d_r}$ is a known matrix with full column rank. The model involves two types of deterministic parameters: $\mathbf{z}_r\in \mathbb{R}^{d_r}$ is an unknown signal and $\boldsymbol{n}$ is a noise vector distributed as
\begin{equation}
    \boldsymbol{n} \sim \mathcal{N}({\mathbf{0}}, \boldsymbol{\Sigma}(\mathbf{z}_n))
\end{equation}
where $\mathbf{z}_n\in \mathbb{R}^{d_n}$ is a nuisance parameter. The goal is to test the presence of a non-zero signal $\mathbf{z}_r$:
\begin{align}\label{linear model33}
    &y=0:\quad \mathbf{z}_r=\boldsymbol{0} \nonumber \\
    &y=1:\quad \mathbf{z}_r\neq \boldsymbol{0}.
\end{align}
\end{definition}

Note that the above Gaussian linear model is very general and allows {\textbf{arbitrary noise covariance with any nuisance parameters}}. Computing GLRT in such models is typically intractable and involves the solution of two challenging optimization problems in inference time as detailed in (\ref{GLRT}). In contrast, evaluating CFARnet is straight forward.

To compare GLRT with BayesCFAR, we define the following fictitious priors:
\begin{itemize}
    \item The prior over $\mathbf{z}_r$ is Gaussian: $p^{\rm fake}(\mathbf{z}_r)\sim \mathcal{N}(0,\sigma_r^2\boldsymbol{I})$.
    \item The prior over $\mathbf{z}_n$ is any non-zero density $p^{\rm fake}(\mathbf{z}_n)$.
    \item The prior over $y$ is ${\rm Pr}^{\rm fake}\left(y=0\right)=p_0>0$.
\end{itemize}

To simplify the presentation, we provide two equivalence results.
The first theorem is based on the assumption that $\mathbf{z}_n$ is known and available as a secondary input to the detectors. In this situation, $T_{\rm GLRT}(\boldsymbol{x},\mathbf{z}_n)$ is based on the true $\mathbf{z}_n$ and an ML estimate of $\mathbf{z}_r$. In BayesCFAR, the detector may depend on $\mathbf{z}_n$ but the objective is still averaged with respect to it. This result is non-asymptotic. It is based on the CFAR property of the linear Gaussian GLRT and can be extended to other non-Gaussian settings in which GLRT is CFAR.

\begin{theorem}
Consider the linear Gaussian model with a large enough $\sigma_r^2$ and a {\textbf{known}} $\mathbf{z}_n$, then there exists a threshold $\gamma$ such that GLRT solves BayesCFAR
\end{theorem}

\begin{proof}

The proof contains three main steps: (a) we show that GLRT is equal to the LRT where $\mathbf{z}_r$ is random with a sufficiently large $\sigma_r$ up to a term that depends only on $\mathbf{z}_n$. (b) we show that for a given rate of FPR, both detectors achieve the maximal TPR for any value of $\mathbf{z}_n$. (c) we use this fact to show that the GLRT with the right threshold minimizes the Bayesian 0-1 loss among all the detectors that have constant FPR over all values of $\mathbf{z}_n$.

Step (a):
when $\mathbf{z}_r$ is random and $\mathbf{z}_n$ is known, the problem reduces to a simple hypothesis test
\begin{align}
    &y=0:\quad \boldsymbol{x}\sim \mathcal{N}\left(\mathbf{0}, \boldsymbol{\Sigma}\right) \nonumber\\
    &y=1:\quad \boldsymbol{x}\sim \mathcal{N}\left(\mathbf{0}, \boldsymbol{\Sigma} + \sigma_r^2\H\H^T\right),
\end{align}
where $\boldsymbol{\Sigma}=\boldsymbol{\Sigma}(\mathbf{z}_n)$.
We denote its LRT by PBL (Partially Bayesian LRT)  which is given by (see Appendix \ref{app_lrt}):
\begin{align}\label{pbl}
    T_{\rm PBL}(\boldsymbol{x},\mathbf{z}_n) & = 2{\rm log}\left(\frac{p(\boldsymbol{x}|y=1)}{p(\boldsymbol{x}|y=0)}\right) \nonumber\\
                         &=\boldsymbol{x}^T\boldsymbol{\Sigma}^{-1}\H\left(\H^T\boldsymbol{\Sigma}^{-1}\H+\frac{1}{\sigma_r^2}\boldsymbol{I}\right)^{-1}\H^T\boldsymbol{\Sigma}^{-1}\boldsymbol{x} \nonumber \\
                        & +  {\rm log}\left(\frac{{\rm det}\boldsymbol{\Sigma}}{{\rm det} \left(\boldsymbol{\Sigma} + \sigma_r^2\H\H^T\right)}\right).
\end{align}
Meanwhile, the GLRT is given by (see Appendix \ref{app_glrt}):
\begin{equation}\label{GLRT linera model}
    T_{\rm GLRT}(\boldsymbol{x},\mathbf{z}_n) = \boldsymbol{x}^T\boldsymbol{\Sigma}^{-1}\H\left(\H^T\boldsymbol{\Sigma}^{-1}\H\right)^{-1}\H^T\boldsymbol{\Sigma}^{-1}\boldsymbol{x}.
\end{equation}
Therefore, for large enough $\sigma_r$, the $\frac{1}{\sigma_r^2}$ term  in (\ref{pbl}) vanishes and $\rm PBL$ converges to GLRT up to a term that depends only on $\mathbf{z}_n$:
    \begin{equation}\label{equivaknce up to a term}
    T_{\rm PBL}(\boldsymbol{x},\mathbf{z}_n) = T_{\rm GLRT}(\boldsymbol{x},\mathbf{z}_n) + c(\mathbf{z}_n),
\end{equation}
where \begin{equation}
    c(\mathbf{z}_n) = {\rm log}\left(\frac{{\rm det}\boldsymbol{\Sigma}}{{\rm det} \left(\boldsymbol{\Sigma} + \sigma_r^2\H\H^T\right)}\right).
\end{equation}

This is reminiscent of the ML and JMAP-ML relation in \cite{yeredor2000joint}.

Step (b): we now show that despite the $c(\mathbf{z}_n)$ term, if we constrain the detectors to have an FPR of $\alpha$ for any value $\mathbf{z}_n$, then both detectors and their thresholds are equivalent.
Because $\mathbf{z}_n$ is known, the thresholds $\gamma_{PBL}$ can depend on $\mathbf{z}_n$, and are given implicitly by:
\begin{equation}
    \alpha = \int_{\gamma_{\rm PBL}}^{\infty}p^0_{\rm PBL}(\tau,\mathbf{z}_n)d\tau,
\end{equation}
where $p^0_{\rm PBL}$ is the PDF of $T_{\rm PBL}(\boldsymbol{x},\mathbf{z}_n)$ for a given $\mathbf{z}_n$, under the $y=0$ hypothesis. Similarly, the threshold of the GLRT is given implicitly by:
\begin{equation}
    \alpha = \int_{\gamma_{\rm GLRT}}^{\infty}p^{0}_{\rm GLRT}(\tau)d\tau,
\end{equation}
where $p^{0}_{\rm GLRT}$ is the PDF of $T_{\rm GLRT}(\boldsymbol{x},\mathbf{z}_n)$ for a given $\mathbf{z}_n$, under the $y=0$ hypothesis. As the PDF of the GLRT does not depend on $\mathbf{z}_n$ \anote{page} \cite{kay1998fundamentals}, the threshold also does not depend on $\mathbf{z}_n$. Using (\ref{equivaknce up to a term}), the PDFs of the detectors satisfy:
\begin{equation}
    p^0_{\rm PBL}(\tau-c(\mathbf{z}_n),\mathbf{z}_n) = p^{0}_{\rm GLRT}(\tau)
\end{equation}
and thus
\begin{align}\label{fprs_PBL_glrt}
    \int_{\gamma_{\rm GLRT}}^{\infty}p^{0}_{\rm GLRT}(\tau)d\tau &= \int_{\gamma_{\rm PBL}}^{\infty}p^{0}_{\rm PBL}(\tau-c(\mathbf{z}_n),\mathbf{z}_n)d\tau \nonumber \\
    &= \int_{\gamma_{\rm PBL}+c(\mathbf{z}_n)}^{\infty}p^0_{\rm PBL}(\tau,\mathbf{z}_n)d\tau  .
\end{align}
From (\ref{fprs_PBL_glrt}) it can be seen the thresholds of the two detectors are related by:
\begin{equation}
    \gamma_{\rm GLRT}(\alpha) = \gamma_{\rm PBL}(\alpha,\mathbf{z}_n) +c(\mathbf{z}_n).
\end{equation}
Thus, by the Neyman-Pearson Lemma, PBL, and thus also the identical GLRT, maximize the TPR for any FPR $\alpha$ and any nuisance parameter $\mathbf{z}_n$.

Step (c): it remains to show that GLRT with the right threshold solves BayesCFAR.
We denote the FPR of a test $(T(\boldsymbol{x},\mathbf{z}_n),\gamma)$ for a given $\mathbf{z}_n$ by:
\begin{equation}
        \alpha(T, \gamma,\mathbf{z}_n)={\rm Pr}\left(T(\boldsymbol{x},\mathbf{z}_n)>\gamma|y=0,\mathbf{z}_n\right),
\end{equation}
and similarly the false negative rate (which is 1-TPR):
\begin{equation}
     \beta(T, \gamma, \mathbf{z}_n)={\rm Pr}\left(T(\boldsymbol{x},\mathbf{z}_n)<\gamma|y=1,\mathbf{z}_n\right).
\end{equation}
The objective of BayesCFAR is therefore:
\begin{equation}
        L_{0-1}(T, \gamma) =p_0\EE{\alpha(T, \gamma, \mathbf{z}_n)} +p_1\EE{\beta(T, \gamma, \mathbf{z}_n)},
\end{equation}
where the expectations are over $p^{\rm fake}(\mathbf{z}_n)$.

Now we show that the GLRT with a threshold $\gamma$ that is generally a function of $\H$ and the priors over $\mathbf{z}_n$ and $y$, is a solution (\ref{CFAR optimization}) which can be written as:
\begin{align}
    {\rm min}_{T(\boldsymbol{x},\mathbf{z}_n),\gamma} \quad & L_{0-1}(T,\gamma) \nonumber \\
    {\rm{s.t.}} \quad & \alpha(T, \tilde \gamma,\mathbf{z}_n)=\alpha(T, \tilde \gamma,\mathbf{z}_n') \quad \forall {\mathbf{z}_n, \mathbf{z}_n',\tilde \gamma},
\end{align}
Note that the constraint must be satisfied for all $\tilde \gamma$ and not just for the optimal $\gamma$.

Due to the CFAR constraint, the FPR is constant with respect to  $\mathbf{z}_n$ and so is its expectation:
\begin{equation}
    \EE{\alpha(T,\gamma, \mathbf{z}_n)}=\alpha(T,\gamma),
\end{equation}
where $\alpha(T,\gamma)$ is the FPR of the test on any value of  $\mathbf{z}_n$. Thus the Bayesian 0-1 loss can be written as:
\begin{equation}\label{Bayesian 0-1 loss}
    L_{0-1}(T, \gamma) =p_0\alpha(T,\gamma) + p_1\EE{\beta(T,\gamma,\mathbf{z}_n))}.
\end{equation}
The best threshold for any detector can be therefore found by minimizing (\ref{Bayesian 0-1 loss}) with respect to $\gamma$.
Specifically we denote the optimal threshold of the GLRT detector by $\gamma^*$, and denote the corresponding FPR  as $\alpha^*$.

Now we prove that any other CFAR detector gives a larger or equal Bayesian 0-1 loss.
We assume that there exist a detector $(T'(\boldsymbol{x},\mathbf{z}_n),\gamma')$ that has FPR of $\alpha'$ for any value of $\mathbf{z}_n$. Its Bayesian loss is given by:
\begin{align}
   L_{0-1}(T', \gamma') &= p_0\alpha'+p_1\EE{\beta(T', \gamma', \mathbf{z}_n)} \nonumber\\
            & \geq p_0\alpha'+p_1\EE{\beta(T_{\rm GLRT}, \gamma'_{\rm GLRT}, \mathbf{z}_n)} \nonumber \\
            &=L_{0-1}(T_{\rm GLRT}, \gamma'_{\rm GLRT}) \nonumber \\
            &\geq L_{0-1}(T_{\rm GLRT}, \gamma^*),
\end{align}
where $\gamma'_{\rm GLRT}$ is the threshold that gives FPR of $\alpha'$ to the GLRT. The first inequality is due to the optimality of GLRT among detectors that have FPR $\alpha'$ for any value of $\mathbf{z}_n$. The second inequality is due to the optimality of the threshold $\gamma^*$ for the GLRT detector.

In conclusion, the GLRT with threshold $\gamma^*$ gives the minimum Bayesian loss among all the detectors that have constant false alarm rate over $\mathbf{z}_n$.
\end{proof}

The second theorem considers the more challenging case in which $\mathbf{z}_n$ is unknown. In this case, the GLRT $\tilde{T}_{\rm GLRT}(\boldsymbol{x})$ uses ML to estimate both $\mathbf{z}_r$ and $\mathbf{z}_n$. The result is based on the asymptotic properties of GLRT (but can be extended to non-Gaussian settings).

\begin{theorem}
Consider the linear Gaussian model with a large enough $\sigma_r^2$ and {\textbf{unknown}} $\mathbf{z}_n$, then if $n\gg \max\{d_r,d_n\}$, there exists a threshold $\gamma$ such that GLRT solves BayesCFAR
\end{theorem}

\begin{proof}
Under large data records, GLRT attains its asymptotic distribution
as defined in \cite[(6.24)]{kay1998fundamentals}. When the Fisher information matrix is block diagonal, this distribution is identical to that without (or with known) nuisance parameters as defined in \cite[(6.27)]{kay1998fundamentals}. Thus,
\begin{equation}
    L_{0,1}\left(\tilde{T}_{\rm GLRT}, \gamma^*\right) = L_{0,1}\left(T_{\rm GLRT}, \gamma^*\right),
\end{equation}
and $\tilde{T}_{\rm GLRT}(\boldsymbol{x})$ satisfies the CFAR constraint.
A detector that does not use $\mathbf{z}_n$ as input cannot be better than the optimal detector that uses $\mathbf{z}_n$ as input. Thus, the GLRT with unknown $\mathbf{z}_n$ and the threshold $\gamma^*$ is the minimizer of the Bayesian 0-1 loss among CFAR detectors.

It remains to clarify what exactly are the conditions for large data records. The assumptions in \cite[p. 206]{kay1998fundamentals} are: large data, weak signal and that the MLE attains its asymptotic density. The first and third assumptions are satisfied in the Gaussian case when $n\gg \max\{d_r,d_n\}$. In appendix \ref{app_asy}, we prove that the assumption of weak signal is not necessary in our setting.

\end{proof}

\begin{figure*}[h]
\centering
     \includegraphics[width=0.285\textwidth]{plots/sigma05_Gaussain_2.png}\includegraphics[width=0.285\textwidth]{plots/sigma1_Gaussain_2.png}\includegraphics[width=0.285\textwidth]{plots/fpr_fprGaussain_2.png} \\
     \includegraphics[width=0.285\textwidth]{plots/sigma05_non_Gaussain_2.png}\includegraphics[width=0.285\textwidth]{plots/sigma1_non_Gaussain_2.png}\includegraphics[width=0.285\textwidth]{plots/fpr_fprnon_Gaussain_2.png}
\caption{Performance graphs in terms of FPR, TPR and thresholds for $\sigma\in\{0,1\}$. Top row is with Gaussian noise and indeed the Gaussian GLRT is best both in terms of ROC and CFAR. NET succeeds in achieving its ROC but is not CFAR. CFARnet is both accurate and CFAR. Bottom row is with non-Gaussian noise and the learned detectors beat the Gaussian GLRT in accuracy. CFARnet is near CFAR with a slight decrease in accuracy, but again is approximately CFAR }

\label{fig:roc_unkown}%
\end{figure*}

The theorems give more insight on CFARnet as a deep learning approximation of BayesCFAR. In the asymptotic Gaussian setting, GLRT is CFAR and there is no degradation in ROC performance due to the additional constraint. This holds for Gaussian models with arbitrary covariance structures. More generally, there is an inherent tradeoff between accuracy and CFAR and a sweet spot must be identified for each application. The theorems also gives a new interpretation to the GLRT, Wald  and Rao tests, that have the same asymptotic distributions \cite[p. 205]{kay1998fundamentals} but do not directly minimize any cost function. Apparently, in asymptotic Gaussian linear models, they are the minimizers of the Bayesian 0-1 loss subject to the CFAR constraint.

\section{Numerical experiments}

In this section, we demonstrate the advantages of CFARnet via numerical experiments.

\subsection{Signal in uncorrelated noise}

In our first experiment, we consider the classical Example 1 as defined in (\ref{DC})-(\ref{params}).
We compare three detectors:
\begin{itemize}
    \item GLRT: Assuming Gaussian noise, the classical GLRT has a simple closed form solution $T_{\rm{GLRT}}=(\boldsymbol{x}^T\mathbf{1})^2/(\boldsymbol{x}^T\boldsymbol{x})$ and is known to be CFAR.

    \item NET: A learned neural network as in Algorithm 1. We choose a uniform fake prior for the unknown parameters in (\ref{params}).  The architecture is based on four  non-linear features: the sample mean of $\boldsymbol{x}$, its sample variance and robust versions of the two based on the median. These features are passed through a fully connected neural network, and are optimized to minimize a cross entropy loss using PyTorch.
    \item CFARnet: A learned neural network as in Algorithm 2. Architecture and implementation are all identical to NET. Loss is cross entropy with an MMD CFAR penalty with parameter $\alpha=1$.
\end{itemize}

The first experiment considers Gaussian noise. In the first row of Fig. 1, we plot the two ROCs for different values of $\sigma$. To examine the CFAR property, we also plot the FPRs under different parameters. As expected, it is easy to see that the Gaussian GLRT performs well and is CFAR. NET provides similar accuracy as illustrated in its ROC but is non-CFAR and results in significantly different FPR when we change $\sigma$. On the other hand, CFARnet is both accurate and near CFAR.

The next setup is more challenging and considers non-Gaussian noise. The setting is identical as before except for the noise distribution
\begin{align}
    p(n_k)= (1-\epsilon)N(0,1) + \epsilon N(0,100)
\end{align}
where $\epsilon=0.1$. This setting represents a Gaussian noise with outliers. There is no simple GLRT for this setting. The results are provided in the second row of Fig. 1. In this case, the Gaussian GLRT is no longer optimal and the two learned detectors provide a significantly better ROC. In terms of CFAR, GLRT is still invariant to the nuisance parameter, but the FPR of NET is dependent on its value. As argued, CFARnet is both accurate and CFAR.

\begin{figure*}[h]
\centering
     \includegraphics[width=0.285\textwidth]{plots/roc_1.png}\includegraphics[width=0.285\textwidth]{plots/roc_2.png}\includegraphics[width=0.285\textwidth]{plots/fpr_fpr.png} \\
\caption{Performance graphs in terms of FPR, TPR and thresholds for two different values of $\boldsymbol{\Sigma}$.
LAMF, NET and CFARnet are better than AMF (especially at the interesting low-FPR regime). Unlike CFARnet, the LAMF and NET are highly non-CFAR.}

\label{fig:aux}%
\end{figure*}

\begin{figure}[h]
\centering
     \includegraphics[width=0.24\textwidth]{plots/gray.png}\hfill\includegraphics[width=0.24\textwidth]{plots/labels.png}
\caption{Illustration of Pavia University dataset. Left is a grayscale image that was generated using the sum of all the channels an in right is a map of the materials annotation.}

\label{fig:paviaU}%
\end{figure}

\begin{figure*}[h]
\centering
     \includegraphics[width=0.285\textwidth]{plots/roc_curve_2.png}\includegraphics[width=0.285\textwidth]{plots/fpr_treshold_net.png}\includegraphics[width=0.285\textwidth]{plots/fpr_threshold_cfarnet.png} \\
\caption{Performance graphs in terms of FPR, TPR and threshold. The left plot shows the ROC curve for one of the materials. The two right plots shows the FPR as a function of the threshold for in each material. While NET is better in terms of ROC curve, it is far from CFAR, while CFARnet is approximately CFAR}
\label{fig:hyper}%
\end{figure*}

\subsection{Gaussian signal in correlated noise}
In our second experiment, we consider the detection of a known signal $\mathbf{s}$ with unknown amplitude in Gaussian noise with unknown covariance $\boldsymbol{\Sigma}$. This is a classical problem in adaptive target detection \cite{kelly1986adaptive, robey1992cfar}. Following these works, we assume a secondary data of $n$ i.i.d. noise-only samples $\boldsymbol{x}_{aux}=(\mathbf{w}_1,...,\mathbf{w}_n)$. Together the observations can be modelled as
\begin{align}
    & \boldsymbol{x} = A\mathbf{s} + \mathbf{w}_0\nonumber\\
    &\boldsymbol{x}_i = \mathbf{w}_i \quad i=1,\cdots, n
\end{align}
where
\begin{equation}
        \mathbf{w}_0,\mathbf{w}_i\sim {\mathcal{N}}({\mathbf{0}}, \boldsymbol{\Sigma})
\end{equation}
The vector $\mathbf{z}$ includes both $A$ and all the elements in the matrix $\boldsymbol{\Sigma}$. The goal is to decide between
\begin{align}
    &y=0:\quad A=0\nonumber \\
    &y=1:\quad A\neq 0.
\end{align}

The  classical adaptive matched filter (AMF) detector is GLRT where the unknown covariance matrix is replaced by its sample covariance \cite{robey1992cfar}:
\begin{equation}\label{ad_glrt}
    T_{\rm AMF}(\boldsymbol{x}) = \frac{\left(\mathbf{s}^T \hat \boldsymbol{\Sigma}^{-1}\boldsymbol{x}\right)^2}{\mathbf{s}^T\hat \boldsymbol{\Sigma}^{-1}\mathbf{s}}
\end{equation}
with
\begin{align}
    \hat\boldsymbol{\Sigma} = \frac 1n \sum_{i=1}^n \mathbf{w}_i\mathbf{w}_i^T.
\end{align}
See also the related detector due to Kelly \cite{kelly1986adaptive}:
\begin{equation}
    T_{\rm Kelly}(\boldsymbol{x}) = \frac{\left(\mathbf{s}^T \hat \boldsymbol{\Sigma}^{-1}\boldsymbol{x}\right)^2}{(\mathbf{s}^T\hat \boldsymbol{\Sigma}^{-1}\mathbf{s})(1+\boldsymbol{x}^T\boldsymbol{\Sigma}^{-1}\boldsymbol{x})}.
\end{equation}

The AMF in (\ref{ad_glrt}) is CFAR but is sub-optimal when $n$ is small. In such settings, it is preferable to plug in regularized covariance estimators:
\begin{equation}
    \hat \boldsymbol{\Sigma}_{\lambda} = \hat \boldsymbol{\Sigma} +\lambda\boldsymbol{I}.
\end{equation}
This leads to the popular diagonally loaded LAMF which is generally not CFAR \cite{ledoit2004well,abramovich2007modified}.

To design a CFAR version of the above detectors, we train a fully connected model with a single hidden layer with the following features as inputs:
\begin{align}
    f^\lambda_1(\boldsymbol{x}) &=\mathbf{s}^T\hat \boldsymbol{\Sigma}_{\lambda}^{-1}\boldsymbol{x} \nonumber \\
    f^\lambda_2(\boldsymbol{x}) &=\mathbf{s}^T\hat \boldsymbol{\Sigma}_{\lambda}^{-1}\mathbf{s} \nonumber \\
    f^\lambda_3(\boldsymbol{x}) &=\boldsymbol{x}^T\hat \boldsymbol{\Sigma}_{\lambda}^{-1}\boldsymbol{x}.
\end{align}
We use 10 different values of  $\lambda$ between 0 and 0.3. The input to the neural network in this case is a concatenation of all 30 features to a single vector. Fig. \ref{fig:aux} shows the ROC curves of AMF, LAMF with $\lambda=0.03$, NET and CFARnet on two different covariances. We also plot the FPRs under different parameters. The performance of LAMF, NET and CFARnet are better than AMF (especially at the interesting low-FPR regime). While LAMF and NET are far from CFAR, CFARnet is approximately CFAR.

\subsection{Target detection in real-world hyperspectral image}
The main motivation to CFARnet is training detectors using a synthetic dataset with continuous large scale parameters. For completeness, our last experiment considers the use of CFARnet in a (semi) real world setting with a finite valued nuisance parameter.
Specifically, we consider target detection in real hyperspectral images annotated according to a finite number of materials. The image is taken from the Pavia University dataset (Fig. \ref{fig:paviaU}).
Fifth of the pixels are annotated to be one of nine materials, e.g., asphalt or trees. We use these annotations as the unknown $\mathbf{z}$ parameters. Following \cite{ziemann2018machine}, we synthetically plant artificial targets on the real image. The target's spectrum is taken from the USGS Spectral Library \cite{kokaly2017usgs} and is interpolated to fit the 103 bands of the hyperspectral image. It is planted randomly (with probability of a half) in the image using an additive model:
\begin{equation}
    \boldsymbol{x}=\alpha \t+\v
\end{equation}
where $\v$ is the original pixel vector, $\t$ is the target and $\alpha=0.05$ is the amplitude of the target. We use two thirds of the data for training and the rest for testing.

We train a simple fully-connected neural network with two hidden layers. First, we fit NET using a standard cross entropy loss and then using CFARnet. To overcome the imbalance between the number of examples of each material, in each step of the stochastic gradient descent, we randomly choose one material and generate a batch for it.
Fig. \ref{fig:hyper} shows that while NET is better in terms of ROC curve, it is far from CFAR. CFARnet shows an expected degradation in performance but is approximately CFAR. To get additional insight on the price of CFAR, we also consider a natural baseline that uses NET but computes its threshold to guarantee a required FPR constraint on the worst case material. Table \ref{partial_auc} compares the partial AUC of the TPR as a function of the worst case FPR in $(0-0.05)$. It can be seen that in all materials except the unlabeled, CFARnet is better with respect to this metric.

\begin{table}[t]
\caption{Partial AUC in $(0-0.05)$ of TPR as a function of the worst case FPR. In all labeled materials CFARnet outperforms NET.}
\label{sample-table}
\vskip 0.15in
\begin{center}
\begin{sc}
\begin{tabular}{lcr}
\toprule
material & net & CFARnet  \\
\midrule
unlabeled & \textbf{0.49} & 0.47 \\
1 & 0.31 & \textbf{0.38} \\
2 & 0.74 & \textbf{0.77} \\
3 & 0.33 & \textbf{0.35} \\
4 & 0.69 & \textbf{0.73} \\
5 & 0.27 & \textbf{0.34} \\
6 & 0.49 & \textbf{0.53} \\
7 & 0.47 & \textbf{0.72} \\
8 & 0.41 & \textbf{0.49} \\
9 & 0.88 & \textbf{0.9} \\

\bottomrule
\end{tabular}
\end{sc}
\end{center}
\vskip -0.1in
\label{partial_auc}

\end{table}

\section{Discussion and future work}
In recent years, deep neural networks are replacing classical
algorithms for detection in many fields. While deep neural
networks give remarkable improvements in accuracy, they lack important properties as CFAR. In this paper we proposed a method to train CFAR neural networks.

Additional research is required to analyze the theoretical and practical decrease in accuracy due to the CFAR constraint. In addition, we proved that CFARnet converges to GLRT in asymptotic linear Gaussian settings. It will be interesting to explore similar properties in more practical setups. Finally, another important direction for future work is to apply the proposed method on large scale and real world problems.

\section*{Acknowledgment}
The authors would like to thank Yoav Wald for fruitful discussions and helpful insights. This research was partially supported by ISF grant number 2672/21.

\section{Introduction}
This document is a model and instructions for \LaTeX.
Please observe the conference page limits.

\section{Ease of Use}

\subsection{Maintaining the Integrity of the Specifications}

The IEEEtran class file is used to format your paper and style the text. All margins,
column widths, line spaces, and text fonts are prescribed; please do not
alter them. You may note peculiarities. For example, the head margin
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and others are deliberate, using specifications that anticipate your paper
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Please do not revise any of the current designations.

\section{Prepare Your Paper Before Styling}
Before you begin to format your paper, first write and save the content as a
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Keep your text and graphic files separate until after the text has been
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\subsection{Abbreviations and Acronyms}\label{AA}
Define abbreviations and acronyms the first time they are used in the text,
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\subsection{Units}
\begin{itemize}
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\end{itemize}

\subsection{Equations}
Number equations consecutively. To make your
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\begin{equation}
a+b=\gamma\label{eq}
\end{equation}

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\subsection{\LaTeX-Specific Advice}

Please use ``soft'' (e.g., \verb|\eqref{Eq}|) cross references instead
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\subsection{Some Common Mistakes}\label{SCM}
\begin{itemize}
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\end{itemize}
An excellent style manual for science writers is \cite{b7}.

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\subsection{Identify the Headings}
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Component heads identify the different components of your paper and are not
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\begin{table}[htbp]
\caption{Table Type Styles}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Table}&\multicolumn{3}{|c|}{\textbf{Table Column Head}} \\
\cline{2-4}
\textbf{Head} & \textbf{\textit{Table column subhead}}& \textbf{\textit{Subhead}}& \textbf{\textit{Subhead}} \\
\hline
copy& More table copy$^{\mathrm{a}}$& &  \\
\hline
\multicolumn{4}{l}{$^{\mathrm{a}}$Sample of a Table footnote.}
\end{tabular}
\label{tab1}
\end{center}
\end{table}

\begin{figure}[htbp]
\centerline{\includegraphics{fig1.png}}
\caption{Example of a figure caption.}
\label{fig}
\end{figure}

Figure Labels: Use 8 point Times New Roman for Figure labels. Use words
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\section*{Acknowledgment}

The preferred spelling of the word ``acknowledgment'' in America is without
an ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B.
G.) thanks $\ldots$''. Instead, try ``R. B. G. thanks$\ldots$''. Put sponsor
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\section*{References}

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For papers published in translation journals, please give the English
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redpajama_set_name: RedPajamaArXiv


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Do We Judge Others by Their Hair?
When you see other people whatever colour they may be do you judge them by the way they wear their hair?
Do you think a natural sister is trying to make a statement? Well is she?
Or that someone with a bright red Rhinna style relaxed hair is trying to look white? Well is she?
"Good hair means waves or curls, Bad hair means you look like a slave"
"Does the way I wear my hair make me a better person"
To quite honest I dont really think that much about my hair and really just want to get through the day without having to think about it too much about it. For me the ultimate low maintenance hair stye would be just to shave it off.
I am a bit slow and behind the times and really not down with the kids as I actually hadn't heard this song before. It is a nice catchy tune and has a simple message.
I am not my hair and how I wear it does not define who I really am. So dont judge me!
I guess I just wanted to say that we all need to make our own decisions about how we wear our hair, how we are comfortable with our hair. How do you feel about a girl wearing a mini skirt getting unwanted attention from men, should she cover herself up? Isn't that the same as saying a black man with cornrows will attract attention from the police, shouldn't he just shave it off. Think about it.
The whole argument reminded me of a case on 'Eli Stone' a smart prime time lawyery type drama on channel five. One of the main characters was representing a friend that was being sued by a young african american lawyer who he was not hired because he was black. This friend was also african american and had a problem with the fact that the younger lawyer had not conformed to white corporate america the way that he had had to. He mostly objected to his unkempt afro.
Most people get from this that the younger lawyer shouldn't have to conform to fit into the older lawyers world. What I get from it is live and let live. The younger lawyer shouldn't judge this older man either as he had his reasons and experiences for who he was and is now and his decisions. Likewise the older lawyer needed to understand that this young man was an outstanding lawyer and the way he wore his hair shouldn't even feature and doesn't need to anymore. I suppose that is what prejudice is all about really, our individual experiences.
Obviously our individual preferences about how we wear our hair are affected by those experiences and do affect us deeply... more than we probably even realise!
I was really excited to get some bentonite as I had heard great things about it. I have quite bad eczema so I wanted to apply this to my skin and hair. I had no idea how bizarre the use of this product would be.
Bentonite Clay or 'healing clay' has the ability to absorb toxins, impurities, heavy metals and other internal contaminants. It is renowned as one of the most powerful and effective healing clays. It can be used internally or externally and has many varied benefits.
Cleanse and detoxify hair and skin with 100% natural bentonite. Reveal lustrous hair and revived skin, let nature take care of you.
The instruction said to mix with water until you get a creamy paste and apply to skin and hair for 15 - 20 minutes. So I mixed a generous amount of paste and slobbered over my hair quickly. Now I suppose my first mistake was just globing it over my hair. It was really thick and after I had lathered over one side of my hair it seemed to stick together and kind of harden. So I would definitely part my hair in six before applying this next time. Once it was on my hair I decided to do the ironing while it was on my hair. After about 10 minutes I was just getting into the ironing groove when my husband walked in. It took a little while to explain why I had covered my hair and body in mud (... oh sorry volcanic ash) and was standing in the middle of our bedroom with mud (... i mean volcanic ash) all over my head, legs and arm, but thats a story for another day.
I decided not to shampoo as I thought this would be counterproductive and I plan to to a protein deep condition in two days time so I thought I would shampoo then. My roots did feel really nice and my hair wasn't tangled even after having all that thick mess on it. A very messy process all in all. I did not get my lustros-o-meter out to check if there was much difference before and after but my hair did feel good afterwards. Just not sure about just washing the dirt (... I mean volcanic detoxifying ash) out with just water. My major problem was the mess in the bathroom after I was finished. All that mud and dirt just got everywhere and I will have to be more careful next time. We are all currently obsessed by natural products. I dont know about you, but I hate the way my bathroom smells of egg after I have used this raw ingredient to condition my hair.
What was truly amazing was my hair the next day. After I finished washing my hair I did my usual leave in, Jane Carter Wrap and Roll and Revitalising Leave In and then air dried, then plaited in one ponytail when dry and went to bed. Nothing special, no blow drying or hot combing. The next day after I let the plait out, I really noticed how straight my hair was and how thick and full it look. I know it wasn't just the Jane Carter stuff as I have been using those for six weeks and my hair hadn't been this full and bouncy. To be quite honest I dont think my hair has ever been this bouncy. I really liked the way the fullness lasted all day.
I didn't find much difference with my skin. The usual itching that night and the next day. Not sure if I have to persevere, but I cant say it worked very well on my skin.
Definitely a keeper, I wanted to use this for its detox properties, but i also loved the way my hair felt the next day, I truly wasn't expecting all that bounce. Me Likey - 4 out of 5.


Stellar's values attract the best and brightest staff; who strive to succeed in every way – whether it's to hit their KPIs or to support the causes that mean the most to them.
Stellar is proud to support a number of charities and organisations at our centre level, raising funds for a variety of programs. Centre managers work with their teams to identify the local causes that mean the most to them; and our staff dress up, bake cupcakes, and pound the pavement to do their part to improve their local communities.
At a corporate level, all of the Stellar staff come together to support some key charities that are near to our hearts.
CEO Melissa Hamilton is passionate about advocating for homelessness, and Stellar is behind her every year for the CEO Sleepout. Vinnie's CEO Sleepout invites CEOs every year to sleep rough for one night, raising funds for vulnerable individuals.
Melissa has participated four times, the most recent being in 2017. Find out more about Stellar's fundraising efforts here.
The Big Issue is an independent, not-for-profit organisation that creates jobs for homeless, marginalised and disadvantaged people. It runs social enterprises to create employment for people unable to access mainstream jobs.
Each of Stellar's centres in Australia is subscribed to The Big Issue as part of The Big 100 program, helping The Big Issue to employ up to 10 disadvantaged women to pack magazines for dispatch.
Save the Children is one of Australia's largest aid and development agencies, dedicated to helping children. They support millions of the hardest-to-reach and most vulnerable children and young people around the world.
Stellar is proud to be sponsoring 50 children, their families and their community in Camanava in the Philippines. We send letters and care parcels to the children regularly and hold bake sales to raise extra funds.
Our fundraising supports practical measures, such as purchasing literacy kits and school supplies for our sponsor children, as well as local projects around sanitation and water supplies.
Stellar is proud to support these deserving causes, and we share news about our initiatives on our social media pages.
Follow us on Twitter or Facebook to discover more.


'Yes,' I said. 'For a six days.' With that I placed my bag in the boot of my car and waved him a cheery farewell as I set off to join eleven other 'Hermits in Company' who had also chosen, for a short space of time, to put aside computers, mobile phones, televisions and radios, and the mad pace of modern life.
Hermits in Company is facilitated by Jane, a psychotherapist and Alexander Technique teacher, as part of the Bede Griffiths Sangha which meets regularly during the year to provide a place for spiritual reflection and growth. Father Bede Griffiths, who died in 1993, was one of the great spiritual teachers of the 20th century, and wrote numerous books on spiritual awakening and the power of Love.
Jane, who has been a member of the Bede Griffiths Sangha for many years, is commited to offering people space for contemplative silence and reflection – not alone but in community.
I understand that. Several years ago I went on an eight day Ignatius retreat at St Beunos, North Wales. My only contact with anyone was half an hour in the morning with my spiritual mentor. Otherwise, the time was my own. I seem to remember spending a lot of it asleep or visiting beautiful overgrown holy wells that mark ancient pilgrim routes in a craggy landscape where sheep now quietly munch. I enjoyed my time and got a huge amount of out of it, but I felt very much alone.
I discovered the existence of Hermits through Jane after my life fell apart a little over three years ago. I believe those precious few silent days saved me. Being in silence, but also in company, gave me the chance to grieve to the depth of my soul with people around, but without anyone being able telling me what I should do or should think. I have been a committed member of Hermits ever since.
The glory of Hermits in Company, which very loosely follows the Benedictine routine of an hour of mediation and prayers before breakfast, lunch, and supper is that it provides a real sense of community amidst the silence that we all long for.
We come together at Michaelgarth, a wonderfully ramshackled, dusty old house fossilised in the 1930s, and set in the glorious grounds of The Society of the Sacred Cross, Tymawr Covent, near Monmouth. It is home to six extraordinary Anglican nuns who have dedicated their lives to prayer, solitude, and silence. The oldest Sister is ninety-four! (They welcome retreatants, visitors, and self-catering groups. Please visit their website for more information).
Some of the Hermits are Christians. Some of us lean towards Buddhism, Hinduism, or other Eastern teachings. I don't know how to describe my faith. It's a bit of everything that makes sense to me. But no-one pushes their beliefs. The retreat is about exploring and deepening whatever belief or faith you have for yourself.
While my Hermit cohorts are walking, or sitting quietly, or doing whatever they want and need to do, my personal task is to run the kitchen and provide meals for the group. A task I thoroughly enjoy as chopping veggies and stirring pots in quietude provides a rich time for reflection and contemplation, and I find that eating in companionable silence makes mealtimes truly pleasurable.
The one interruption we have in our daily routine is on Sunday morning, when we are allowed to join the Sisters for communion. It is a delightfully gently and joyous experience that takes place in the chapel, which is part of the convent.
There is something about being in the company of nuns dressed in traditional black habits who are used to singing together that makes the service quite magical, and even mystical. Listening to them always ignites a primordial memory in me, from who knows when, and this is the only occassion I ever feel moved to receive the Eucharist.
But this time, a day or so later, there is another enchanting interruption to our routine. As this is early spring we are invited to join the Sisters in the Rogationtide Prayers for the convent. They open their arms in welcome to us without care for what our beliefs may be.
We gather together at the back of the convent and proceed in a hushed giggling gaggle around the grounds blessing the two orchards, the meadow, the garden, the neighbours, the machinery, the water supply, the house, computers, broadband, and washing machines (!), the kitchen garden, and the flowers in the beautiful Lady Garden. I feel as if I have stepped back to the turn of the nineteenth century when these sacred country rituals were part of daily life, enriching us and reminding us of the preciousness of nature.
Shame on us that we have all but for forgotten.
I always find it difficult to relinquish the silence as the retreat draws to an end. Rather, I feel I am just settling into the miraculous abundance that Michaelgarth and the convent offers, which has nothing to do with money, or power, or possessions. But modern life calls again to us all, and so we must return to it.
'Were you really silent for a week?' asked my boater neighbour upon my return to the marina, puffing on his E-cigarette.
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