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Functions are Vectors This is a great blog post from Max Slater about how viewing functions as infinite dimensional vectors unlocks powerful tools from linear algebra. I think this is an under-emphasized perspective in intro linear algebra classes.

Alec Helbling

10,765 subscribers

93,507 次观看 • 9 个月前 •via X (Twitter)

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"great design is invisible" when explaining this concept to non-design minded customers, I often show them this from Linear. it's just a wonderful example of thinking

"great design is invisible" when explaining this concept to non-design minded customers, I often show them this from Linear. it's just a wonderful example of thinking

Arron ✸

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Ever seen a chord that swings both ways? In this excerpt from Schubert, a seventh chord [F-A-C-E♭] first functions as a dominant of B♭, then is respelled with a D♯ instead of E♭ and functions as a German Augmented Sixth chord in A minor.

Ever seen a chord that swings both ways? In this excerpt from Schubert, a seventh chord [F-A-C-E♭] first functions as a dominant of B♭, then is respelled with a D♯ instead of E♭ and functions as a German Augmented Sixth chord in A minor.

Robert Komaniecki

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I think this is a great idea… from America’s 250th anniversary… Take a road trip and see how wonderful our country is. A great idea from Secretary Sean Duffy and President Trump.

I think this is a great idea… from America’s 250th anniversary… Take a road trip and see how wonderful our country is. A great idea from Secretary Sean Duffy and President Trump.

Tim Young

52,450 次观看 • 3 个月前

Full Fine-tuning vs. Freezing Layers. Interact 👉 and == Full Fine-tuning == A real network has many — three layers in this example, billions of parameters in a production model. What does fine-tuning look like when you update all of them? That’s full fine-tuning: continue training every weight in the pretrained network on your new task. Every layer’s W gets its own ΔW. Nothing is frozen — every parameter is in play. Think of an MLP as a chain of prerequisites leading to an advanced course. Layer 1 might be Linear Algebra, layer 2 Probability, layer 3 Advanced Machine Learning — each one building on what came before. Fine-tuning is what happens during graduate study: the foundations are already there from undergrad, so you’re not re-learning. Full fine-tuning is reviewing every prerequisite to see what new topics have appeared and what discoveries the field has made since the last time you sat through them. Effective — but exhausting. This diagram shows the same three-layer MLP twice, side by side. On the left, the pretrained network runs on input X: three weight matrices W₁, W₂, W₃, each followed by a ReLU activation. Full fine-tuning gives the model the most freedom to specialize. Every parameter can move — and every parameter that can move must be stored. But not every prerequisite needs revisiting. The further you go back in the chain, the less the material has changed since pretraining — the linear-algebra basics under your computer-vision course are largely the same as they ever were. The next page does exactly that: freeze the prerequisites that haven’t moved, and only refresh the advanced one closest to your specialization. == Freezing Layers == Full fine-tuning reviewed every prerequisite — Linear Algebra, Probability, Advanced ML — to refresh each subject with the latest topics. Effective, but exhausting. Then you realize something. The prerequisites haven’t actually changed that much. Linear Algebra is still Linear Algebra; the matrix decompositions you learned still hold. Probability is still Probability; the distributions and Bayes’ rule haven’t moved. Almost all the new material — the new ideas, the recent discoveries — lives in the advanced layer at the top. That’s freezing layers: keep the prerequisite layers fixed at their pretrained state, and only update the advanced one. In the diagram below, W1 and W2 — the foundational prerequisites — stay frozen. Only W3 — the layer closest to your task-specific output — gets a ΔW.

Full Fine-tuning vs. Freezing Layers. Interact 👉 and == Full Fine-tuning == A real network has many — three layers in this example, billions of parameters in a production model. What does fine-tuning look like when you update all of them? That’s full fine-tuning: continue training every weight in the pretrained network on your new task. Every layer’s W gets its own ΔW. Nothing is frozen — every parameter is in play. Think of an MLP as a chain of prerequisites leading to an advanced course. Layer 1 might be Linear Algebra, layer 2 Probability, layer 3 Advanced Machine Learning — each one building on what came before. Fine-tuning is what happens during graduate study: the foundations are already there from undergrad, so you’re not re-learning. Full fine-tuning is reviewing every prerequisite to see what new topics have appeared and what discoveries the field has made since the last time you sat through them. Effective — but exhausting. This diagram shows the same three-layer MLP twice, side by side. On the left, the pretrained network runs on input X: three weight matrices W₁, W₂, W₃, each followed by a ReLU activation. Full fine-tuning gives the model the most freedom to specialize. Every parameter can move — and every parameter that can move must be stored. But not every prerequisite needs revisiting. The further you go back in the chain, the less the material has changed since pretraining — the linear-algebra basics under your computer-vision course are largely the same as they ever were. The next page does exactly that: freeze the prerequisites that haven’t moved, and only refresh the advanced one closest to your specialization. == Freezing Layers == Full fine-tuning reviewed every prerequisite — Linear Algebra, Probability, Advanced ML — to refresh each subject with the latest topics. Effective, but exhausting. Then you realize something. The prerequisites haven’t actually changed that much. Linear Algebra is still Linear Algebra; the matrix decompositions you learned still hold. Probability is still Probability; the distributions and Bayes’ rule haven’t moved. Almost all the new material — the new ideas, the recent discoveries — lives in the advanced layer at the top. That’s freezing layers: keep the prerequisite layers fixed at their pretrained state, and only update the advanced one. In the diagram below, W1 and W2 — the foundational prerequisites — stay frozen. Only W3 — the layer closest to your task-specific output — gets a ΔW.

Tom Yeh

27,587 次观看 • 4 个月前

Vector Database by Hand ✍️ Vector databases are revolutionizing how we search and analyze complex data. They have become the backbone of Retrieval Augmented Generation (#RAG). How do vector databases work? [1] Given ↳ A dataset of three sentences, each has 3 words (or tokens) ↳ In practice, a dataset may contain millions or billions of sentences. The max number of tokens may be tens of thousands (e.g., 32,768 mistral-7b). Process "how are you" [2] 🟨 Word Embeddings ↳ For each word, look up corresponding word embedding vector from a table of 22 vectors, where 22 is the vocabulary size. ↳ In practice, the vocabulary size can be tens of thousands. The word embedding dimensions are in the thousands (e.g., 1024, 4096) [3] 🟩 Encoding ↳ Feed the sequence of word embeddings to an encoder to obtain a sequence of feature vectors, one per word. ↳ Here, the encoder is a simple one layer perceptron (linear layer + ReLU) ↳ In practice, the encoder is a transformer or one of its many variants. [4] 🟩 Mean Pooling ↳ Merge the sequence of feature vectors into a single vector using "mean pooling" which is to average across the columns. ↳ The result is a single vector. We often call it "text embeddings" or "sentence embeddings." ↳ Other pooling techniques are possible, such as CLS. But mean pooling is the most common. [5] 🟦 Indexing ↳ Reduce the dimensions of the text embedding vector by a projection matrix. The reduction rate is 50% (4->2). ↳ In practice, the values in this projection matrix is much more random. ↳ The purpose is similar to that of hashing, which is to obtain a short representation to allow faster comparison and retrieval. ↳ The resulting dimension-reduced index vector is saved in the vector storage. [6] Process "who are you" ↳ Repeat [2]-[5] [7] Process "who am I" ↳ Repeat [2]-[5] Now we have indexed our dataset in the vector database. [8] 🟥 Query: "am I you" ↳ Repeat [2]-[5] ↳ The result is a 2-d query vector. [9] 🟥 Dot Products ↳ Take dot product between the query vector and database vectors. They are all 2-d. ↳ The purpose is to use dot product to estimate similarity. ↳ By transposing the query vector, this step becomes a matrix multiplication. [10] 🟥 Nearest Neighbor ↳ Find the largest dot product by linear scan. ↳ The sentence with the highest dot product is "who am I" ↳ In practice, because scanning billions of vectors is slow, we use an Approximate Nearest Neighbor (ANN) algorithm like the Hierarchical Navigable Small Worlds (HNSW).

Vector Database by Hand ✍️ Vector databases are revolutionizing how we search and analyze complex data. They have become the backbone of Retrieval Augmented Generation (#RAG). How do vector databases work? [1] Given ↳ A dataset of three sentences, each has 3 words (or tokens) ↳ In practice, a dataset may contain millions or billions of sentences. The max number of tokens may be tens of thousands (e.g., 32,768 mistral-7b). Process "how are you" [2] 🟨 Word Embeddings ↳ For each word, look up corresponding word embedding vector from a table of 22 vectors, where 22 is the vocabulary size. ↳ In practice, the vocabulary size can be tens of thousands. The word embedding dimensions are in the thousands (e.g., 1024, 4096) [3] 🟩 Encoding ↳ Feed the sequence of word embeddings to an encoder to obtain a sequence of feature vectors, one per word. ↳ Here, the encoder is a simple one layer perceptron (linear layer + ReLU) ↳ In practice, the encoder is a transformer or one of its many variants. [4] 🟩 Mean Pooling ↳ Merge the sequence of feature vectors into a single vector using "mean pooling" which is to average across the columns. ↳ The result is a single vector. We often call it "text embeddings" or "sentence embeddings." ↳ Other pooling techniques are possible, such as CLS. But mean pooling is the most common. [5] 🟦 Indexing ↳ Reduce the dimensions of the text embedding vector by a projection matrix. The reduction rate is 50% (4->2). ↳ In practice, the values in this projection matrix is much more random. ↳ The purpose is similar to that of hashing, which is to obtain a short representation to allow faster comparison and retrieval. ↳ The resulting dimension-reduced index vector is saved in the vector storage. [6] Process "who are you" ↳ Repeat [2]-[5] [7] Process "who am I" ↳ Repeat [2]-[5] Now we have indexed our dataset in the vector database. [8] 🟥 Query: "am I you" ↳ Repeat [2]-[5] ↳ The result is a 2-d query vector. [9] 🟥 Dot Products ↳ Take dot product between the query vector and database vectors. They are all 2-d. ↳ The purpose is to use dot product to estimate similarity. ↳ By transposing the query vector, this step becomes a matrix multiplication. [10] 🟥 Nearest Neighbor ↳ Find the largest dot product by linear scan. ↳ The sentence with the highest dot product is "who am I" ↳ In practice, because scanning billions of vectors is slow, we use an Approximate Nearest Neighbor (ANN) algorithm like the Hierarchical Navigable Small Worlds (HNSW).

Tom Yeh

192,022 次观看 • 2 年前

One more post from this day 😂 This gentleman watched me catch this fish and came running barefoot in the rain from a few houses over. Genuine interaction, this is what these moments are all about!

One more post from this day 😂 This gentleman watched me catch this fish and came running barefoot in the rain from a few houses over. Genuine interaction, this is what these moments are all about!

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If we saw “Lost Ancient Technology” would we recognize it? This object was pulled from the seabed at the sunken port of Thonis-Heracleion in Egypt. Noted as a scepter the relic displays multiple features of some kind of drive shaft. I could think of multiple mechanical functions for this object.

Ancient Hypotheses

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2 clips from the same event. 1st by an international cameraman, 2nd by ANI ANI is instructed to never show the teleprompter. They posted 10+ long clips from this event in France & all show the same angle This is how insecure & desperate they are to project him as a great orator

2 clips from the same event. 1st by an international cameraman, 2nd by ANI ANI is instructed to never show the teleprompter. They posted 10+ long clips from this event in France & all show the same angle This is how insecure & desperate they are to project him as a great orator

SirKazam {blu tik}

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From my previous joke comic I post about Charlie having fur. I think it's funny people ask if she would fluff up in winter.. This is just how I imagine they would be on a cold morning when no one wants to get up from bed after a sleepless night.😏 #chaggie

From my previous joke comic I post about Charlie having fur. I think it's funny people ask if she would fluff up in winter.. This is just how I imagine they would be on a cold morning when no one wants to get up from bed after a sleepless night.😏 #chaggie

RFakonWolf_Art

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Always a privilege to be part of Canada’s constitutional traditions, especially as Leader of His Majesty’s Loyal Opposition in the Senate. As Canadians, we benefit from a continuity and stability that’s easy to take for granted. The Governor General, as the King’s representative in Canada, is the integral instrument of our constitutional architecture, quietly ensuring continuity, stability, and impartiality in how our system functions. At a time when democratic institutions are under strain around the world, there is value in pausing to recognize how ours endures.

Always a privilege to be part of Canada’s constitutional traditions, especially as Leader of His Majesty’s Loyal Opposition in the Senate. As Canadians, we benefit from a continuity and stability that’s easy to take for granted. The Governor General, as the King’s representative in Canada, is the integral instrument of our constitutional architecture, quietly ensuring continuity, stability, and impartiality in how our system functions. At a time when democratic institutions are under strain around the world, there is value in pausing to recognize how ours endures.

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i dont think you guys understand how impressive this is from a dancer’s perspective, the original trend only did that 360° leg spin move once but jungkook was able to do it twice in a row AND ON BOTH LEGS

i dont think you guys understand how impressive this is from a dancer’s perspective, the original trend only did that 360° leg spin move once but jungkook was able to do it twice in a row AND ON BOTH LEGS

kin

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Ah what a great level this is! I love this level! Its so FUN! Can we please get more verifications from doggie its honestly crazy how fun they are!

Ah what a great level this is! I love this level! Its so FUN! Can we please get more verifications from doggie its honestly crazy how fun they are!

Clare

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When I say Spencer Rattler was in his bag from a sack-evasion perspective against the Giants ... this is the type of rep I'm talking about. I genuinely don't think you can teach this type of thing. You either have it or you don't. #Saints

When I say Spencer Rattler was in his bag from a sack-evasion perspective against the Giants ... this is the type of rep I'm talking about. I genuinely don't think you can teach this type of thing. You either have it or you don't. #Saints

Jeff Nowak

177,612 次观看 • 10 个月前

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🐨: (talking about ‘into the sun’) i think taehyung worked very well with other people. this song was great from the start, so i said we had to put this in, no matter what. jin hyung’s voice in this is very beautiful

hourly namjin

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This is not a scene from a Mad Max movie. This is footage from the funeral of dictator Ali Khamenei in Tehran, Iran, yesterday. Supporters of the Islamic regime, including IRGC terrorists, are every bit as mad and fanatical as shown in this footage. One of them even chained himself to the front of the truck carrying the filthy corpse of the regime’s terrorist supreme leader. This is not Mad Max — this is the madness that unfolded yesterday in Tehran.

Babak Taghvaee - The Crisis Watch

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evan answered this perfectly. this is the mindset of a true perfomer! 👤 i know being on stage as a soloist is different from being on stage in a group. what’s different for you? 🦌 i don’t think anything has really changed. i think what’s more important is how much energy we bring to the stage

evan answered this perfectly. this is the mindset of a true perfomer! 👤 i know being on stage as a soloist is different from being on stage in a group. what’s different for you? 🦌 i don’t think anything has really changed. i think what’s more important is how much energy we bring to the stage

h.

88,944 次观看 • 4 天前

“The best thing about being an apprentice is I get to learn while I earn as well." This #NAW2025, we asked apprentices from OxfordshireAdvancedSkills how they're gaining #SkillsForLife in science & tech roles. Apprentices are vital for businesses & driving innovation.

“The best thing about being an apprentice is I get to learn while I earn as well." This #NAW2025, we asked apprentices from OxfordshireAdvancedSkills how they're gaining #SkillsForLife in science & tech roles. Apprentices are vital for businesses & driving innovation.

Department for Science, Innovation and Technology

14,979 次观看 • 1 年前

Jamie Foxx: "I kill all the white people in the movie, how great is that?" Thunderous applause from the audience Imagine if this was about any other race

Jamie Foxx: "I kill all the white people in the movie, how great is that?" Thunderous applause from the audience Imagine if this was about any other race

End Wokeness

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There is no fall damage in Assassin's Creed Black Flag Resynced. Edward can fall from a great height without taking any damage, making him look as powerful as Kassandra in this clip. This is likely a bug in the preview build and will hopefully be fixed for the launch version.

There is no fall damage in Assassin's Creed Black Flag Resynced. Edward can fall from a great height without taking any damage, making him look as powerful as Kassandra in this clip. This is likely a bug in the preview build and will hopefully be fixed for the launch version.

Zephryss⛩️

93,425 次观看 • 2 个月前