This research paper presents SIMPL (Scalable Iterative Maximization of Population-coded Latents), a novel, computationally efficient algorithm designed to refine the estimation of latent variables and tuning curves from neural population activity. Latent variables in neural data represent essential low-dimensional quantities encoding behavioral or cognitive states, which neuroscientists seek to identify to understand brain computations better. Background and Motivation Traditional approaches commonly assume the observed behavioral variable as the latent neural code. However, this assumption can lead to inaccuracies because neural activity sometimes encodes internal cognitive states differing subtly from observable behavior (e.g., anticipation, mental simulation). Existing latent variable models face challenges such as high computational cost, poor scalability to large datasets, limited expressiveness of tuning models, or difficulties interpreting complex neural network-based functio...
1. Classification Problem
- Definition: Classification
is a supervised learning task where the output variable y
is discrete-valued
rather than continuous.
- In
particular, consider binary
classification where y∈ {0,1} (e.g., spam detection: spam =1, not spam =0).
- Each
training example is a pair (x(i), y(i)), where x(i)∈Rd is a feature vector, and y(i) is the label.
2. Why Not Use Linear Regression for
Classification?
- Linear
regression tries to predict continuous values, which is problematic for
classification as the prediction can be outside [0,1].
- For
example, predicting y≈1.5 or −0.2 is meaningless when y is binary.
- Instead,
we want the output hθ(x) to be interpreted as the probability that y=1
given x.
3. Logistic Regression Model
Hypothesis:
hθ(x)=g(θTx)=1+e−θTx1,
where:
- g(z)=1+e−z1 is the sigmoid function, which
maps any real value to the interval (0, 1).
- θ∈Rd+1 are
parameters (including intercept term).
- hθ(x) can be interpreted as the estimated
probability P(y=1∣x;θ).
Decision Boundary:
- Predict
y=1
if hθ(x)≥0.5; otherwise, predict y=0.
- The
decision boundary corresponds to θTx=0, which is a linear boundary in input
space.
4. Loss Function and Cost Function
Probability Model:
- Logistic
regression models conditional probability directly:
P(y=1∣x;θ)=hθ(x),P(y=0∣x;θ)=1−hθ(x).
- Equivalently,
likelihood for data point (x(i),y(i)):
p(y(i)∣x(i);θ)=(hθ(x(i)))y(i)(1−hθ(x(i)))1−y(i).
Cost (Loss) Function:
- Use negative log-likelihood
(cross-entropy loss) as cost per example:
J(i)(θ)=−[y(i)loghθ(x(i))+(1−y(i))log(1−hθ(x(i)))].
- Overall
cost function (average over n examples):
J(θ)=n1∑i=1nJ(i)(θ).
- This
loss is convex in θ, enabling efficient
optimization.
5. Training Logistic Regression
·
Use
methods such as gradient
descent or more advanced optimization (Newton's method,
quasi-Newton) to minimize cost J(θ).
·
The
gradient of the cost function is:
∇θJ(θ)=n1∑i=1n(hθ(x(i))−y(i))x(i).
- Update rule in gradient descent:
θ:=θ−α∇θJ(θ),
where α is the learning rate.
6. Multi-class Classification
·
When y∈{1,2,...,k} for k>2, logistic regression generalizes to multinomial logistic regression
or Softmax regression.
·
Model
outputs hˉθ(x)∈Rk called logits.
·
The Softmax function converts
logits into probabilities:
P(y=j∣x;θ)=∑s=1kexp(hˉθ(x)s)exp(hˉθ(x)j).
- Loss
for example (x(i),y(i)) is the negative
log-likelihood:
J(i)(θ)=−logP(y(i)∣x(i);θ).
7. Discriminative vs. Generative Classification
Algorithms
- Discriminative
algorithms (like logistic regression) model p(y∣x)
directly or learn a direct mapping from x to y.
- Generative
algorithms model the joint distribution p(x,y)=p(x∣y)p(y).
- Example:
Gaussian Discriminant Analysis (GDA).
- Logistic
regression is an example of a discriminative approach focusing purely on p(y∣x).
8. Linear Hypothesis Class and Decision Boundaries
- Logistic
regression hypothesis class:
H={hθ:hθ(x)=1{θTx≥0}},
which are
classifiers with linear decision boundaries.
- More
generally, hypothesis classes can be extended to neural networks or other
complex architectures.
9. Perceptron Learning as Contrast to Logistic
Regression
·
Perceptron
also uses a linear classifier but with a different loss and update rule.
·
Logistic
regression provides probabilistic outputs and optimizes a convex cost function,
generally yielding better statistical properties.
10. Practical Considerations
- Feature
scaling often improves numerical stability.
- Regularization
(e.g., L2) is frequently added to cost to prevent overfitting.
- Logistic
regression handles input features linearly; non-linear boundaries require
feature engineering or kernel methods.
Summary:
Logistic
regression is a fundamental classification algorithm that models the
conditional probability of the positive class using a sigmoid of a linear
function of input features. It is trained via maximizing likelihood (or
minimizing cross-entropy loss) and extends naturally to multi-class problems
via Softmax. It is a discriminative model focusing directly on p(y∣x) and yields linear decision boundaries. It
contrasts with generative models by its direct approach to classification.
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