Md. Asif Uddin

    Chapter 4 VII.4

    Single-Cell Biology

    A single-cell pipeline is a sequence of decisions, each of which changes the geometry a later clustering will read as biology.

    A pipeline of decisions, each changing the geometry the clustering will read.

    How this chapter is built

    M3Load-bearing

    The content is mathematics. Understanding is demonstrated by computation, not recall.

    basics3/11what the words mean
    concept2/2what to picture
    theory0/4why it works, and when it does not
    mathematics0/15derive it, then compute it
    practice0/9build it, break it, read the papers

    Five strands, not one. Mathematics is the spine; the other four are the body. A chapter cannot pay its way out of teaching with problems, nor out of problems with teaching.

    Before you start

    The problem

    Between the count matrix and the figure lie a dozen choices — filtering, normalisation, feature selection, projection — none of which are reported as results, all of which change them.

    A loss surface, drawn as contoursNested contour rings around a deep minimum on the left and a shallower one on the right, with a saddle between them. A dashed line runs from a starting point downhill into the nearer basin, not necessarily the better one.loss surface over two parametersgloballocalstartThe loss is the only statement of the objective the optimiser can read. What it omits is not optimised.
    Fig. 4 — A loss surface drawn as contours, with a deep basin and a shallower one. Descent finds the nearer minimum, which is not always the better one.

    What this chapter covers

    • scRNA-seq
    • Cells as observations
    • Gene-expression matrices
    • Dimensionality reduction
    • Clustering
    • Cell types

    Apparatus

    The mathematics this chapter leans on, held in Book 0 so it can be assumed here without being taught here. Not a gate — follow a link when a step stops making sense.

    Rank, eigenvalues and the singular value decomposition 0.LA.04 · Lagrange multipliers 0.OP.04 · Multiple comparisons 0.ST.04 · Variance and covariance 0.PR.03

    Notation

    • XA batch of token representations, T×d, rows are tokens
    • dModel width
    • VarVariance
    • λA regularisation coefficient, or an eigenvalue where the context is linear algebra
    • θ̂An estimate, as against the quantity it estimates

    Propositions

    Not yet written. The topics above are the plan for this chapter; each will become a proposition with its own figure.

    Worked problems

    0/5 problems0/4 variants0/10 exercisesowes 15 more

    Not yet written. At M3 this chapter owes 5 worked problems across 4 distinct variants, and 10 exercises, every one with a published solution. The build enforces that from the day the chapter is marked published.